#510517

order	512
nilpotency class	4
isoclinism	isoclinic class of #510517
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31681
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24285	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

571 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 571 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,15,19,10)(4,11)(5,9)(6,14)(7,12)(8,13)(16,43,46,35)(17,33,49,42)(18,34,28,29)(20,38)(21,36)(22,37)(23,30)(24,31)(25,39)(26,32,45,41)(27,40)(44,58,63,57)(47,61,64,56)(48,59,50,54)(51,62)(52,60)(53,55);(2,9)(3,16,19,46)(4,20)(5,7)(6,25)(10,32,15,41)(11,36)(12,13)(17,47,49,64)(18,26,28,45)(21,23)(22,52)(24,27)(29,35,34,43)(31,39)(33,57,42,58)(37,55)(44,50,63,48)(54,56,59,61)(60,62);(1,3)(2,10)(4,17)(5,18)(6,26)(7,28)(8,19)(9,29)(11,33)(12,34)(13,15)(14,41)(16,24)(20,48)(21,49)(22,47)(23,50)(25,46)(27,45)(30,54)(31,35)(32,40)(36,42)(37,61)(38,59)(39,43)(44,53)(51,63)(52,64)(55,58)(56,60)(57,62);(1,4)(2,11)(3,17)(5,23)(6,22)(7,20)(8,21)(9,30)(10,33)(12,38)(13,36)(14,37)(15,42)(16,44)(18,50)(19,49)(24,53)(25,51)(26,47)(27,52)(28,48)(29,54)(31,55)(32,56)(34,59)(35,58)(39,62)(40,60)(41,61)(43,57)(45,64)(46,63)

Polycyclic presentation:

2^9|a^2=efh,b^2=efh,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=defh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gh,h^b=hi,g^c=ghi,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320552, #72320582, #72320806
maximal normal subgroup of	#72320552, #72320582, #72320806
maximal quotient of	#72321036, #72321246, #72321427, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417713);