#6816811

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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

2120 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)(15,20,24,31)(16,26,29,21)(18,22,28,32)(23,35,37,33)(25,40,39,34)(27,36,30,41)(38,43,42,44);(5,6)(7,10)(8,12)(9,11)(14,19)(15,23)(16,27)(18,28)(20,33)(21,36)(24,37)(26,41)(29,30)(31,35)(34,40)(38,42);(1,2,3,4)(13,15,18,23,17,24,28,37)(14,20,22,35,19,31,32,33)(16,25,30,38,29,39,27,42)(21,34,36,44,26,40,41,43);(5,7,9,8)(6,10,11,12)(14,19)(15,24)(21,26)(22,32)(23,37)(25,39)(36,41)(38,42);(5,8,9,7)(6,10,11,12)(13,16,28,27,17,29,18,30)(14,21,32,41,19,26,22,36)(15,25,37,42,24,39,23,38)(20,34,33,43,31,40,35,44)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6724007);