#6794815

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1222 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:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=i,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6702011);