#6433964

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433057
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1632 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6341160);