#6816794

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2157 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=fi,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6723990);