#6704359

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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	#31406	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1320 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:

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

Polycyclic presentation:

2^9|b^2=f,d^2=gi,e^2=fi,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,h^c=hi,h^e=hi

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, 6611555);