#6804837

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1231 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=fgi,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,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, 6712033);