#6909332

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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	#2615	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1525 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=g,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh,e^d=ei

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, 6816528);