#510512

order	512
nilpotency class	4
isoclinism	isoclinic class of #509118
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3199
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#31667
prime radical	#24287	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

554 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 118 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417708);