#511331

order	512
nilpotency class	4
isoclinism	isoclinic class of #511331
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32468
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#525	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 389 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)(16,43,45,35)(17,33,48,42)(18,34,28,29)(20,38)(21,36)(22,37)(23,30)(24,31)(25,39)(26,41,50,32)(27,40)(44,59,64,58)(46,63,53,57)(47,60,49,55)(51,62)(52,61)(54,56);(2,9)(3,16,19,45)(4,20)(5,7)(6,25)(10,32,15,41)(11,36)(12,13)(17,46,48,53)(18,50,28,26)(21,23)(22,52)(24,27)(29,35,34,43)(31,39)(33,58,42,59)(37,56)(44,49,64,47)(55,57,60,63)(61,62);(1,3,5,18,8,19,7,28)(2,10,9,29,13,15,12,34)(4,17,23,49,21,48,20,47)(6,26,24,45,27,50,25,16)(11,33,30,55,36,42,38,60)(14,32,31,43,40,41,39,35)(22,53,54,64,52,46,51,44)(37,57,56,58,61,63,62,59);(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)(18,49)(19,48)(24,54)(25,51)(26,53)(27,52)(28,47)(29,55)(31,56)(32,57)(34,60)(35,59)(39,62)(40,61)(41,63)(43,58)(45,64)(46,50)

Polycyclic presentation:

2^9|a^2=efh,b^2=efh,c^2=ef,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=gh,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, #72320665
maximal normal subgroup of	#72320552, #72320582, #72320665
maximal quotient of	#72320894, #72321080, #72321698, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418527);