#6684921

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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	#39356	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1992 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fi,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,e^d=ei

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, 6592117);