#6686969

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1918 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=g,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,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, 6594165);