#6433755

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433057
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1945 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,24,28)(19,29,20,32)(21,26,25,31)(22,27,23,30);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,26,24,20,22,31)(18,25,27,29,28,21,30,32);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,20,23,31,24,19,22,26)(18,21,27,32,28,25,30,29);(10,14)(13,16)(17,21)(18,26)(19,30)(20,27)(22,29)(23,32)(24,25)(28,31);(17,22,24,23)(18,27,28,30)(19,31,20,26)(21,32,25,29)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,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, 6340951);