#6806900

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#42796	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2015 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,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, 6714096);