#6433949

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ 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) = 128

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

Polycyclic presentation:

2^9|a^2=fi,c^2=g,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,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, 6341145);