#6684935

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682460
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37682	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2048 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(5,6,8,9)(7,11,12,10)(13,14,18,22)(15,24,26,20)(16,28,29,43)(17,23,19,33)(21,30,37,31)(25,39,41,42)(27,36,32,35)(34,40,38,44);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,26)(14,20,22,24)(16,25,29,41)(17,32,19,27)(21,34,37,38)(23,36,33,35)(28,42,43,39)(30,44,31,40);(2,4)(6,9)(10,11)(13,16,17,30,18,29,19,31)(14,21,33,43,22,37,23,28)(15,25,32,44,26,41,27,40)(20,34,35,39,24,38,36,42);(5,7,8,12)(6,10,9,11)(14,22)(15,26)(16,29)(23,33)(27,32)(30,31)(34,38)(39,42);(2,4)(13,17,18,19)(14,23,22,33)(15,27,26,32)(16,30,29,31)(20,35,24,36)(21,28,37,43)(25,40,41,44)(34,39,38,42)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6592131);