#6811522

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2075 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)(7,10)(8,12)(9,11)(13,14,19,18)(15,20,25,33)(16,21,29,28)(17,22,31,32)(23,26,34,39)(24,30,36,43)(27,35,41,42)(37,38,40,44);(5,7,10,6)(8,12,9,11)(13,15,19,25)(14,20,18,33)(16,26,29,39)(17,30,31,43)(21,34,28,23)(22,36,32,24)(27,40,41,37)(35,44,42,38);(2,4)(13,16,14,21,19,29,18,28)(15,23,33,39,25,34,20,26)(17,27,22,35,31,41,32,42)(24,37,43,44,36,40,30,38);(5,8,10,9)(6,11,7,12)(17,31)(22,32)(24,36)(27,41)(30,43)(35,42)(37,40)(38,44);(5,9,10,8)(6,11,7,12)(13,17,18,32,19,31,14,22)(15,24,20,30,25,36,33,43)(16,27,28,42,29,41,21,35)(23,37,26,38,34,40,39,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=hi,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6718718);