#6817284

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	#37680	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgh,d^2=f,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,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, 6724480);