#6435669

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432861
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

1524 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)(11,15)(14,16)(17,18,23,26,24,27,22,25)(19,20,28,29,21,30,31,32);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,24,21)(18,20,27,30)(22,31,23,28)(25,32,26,29);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20)(18,19)(21,27)(22,29)(23,32)(24,30)(25,28)(26,31);(9,11,12,15)(10,14,13,16)(17,21,24,19)(18,20,27,30)(22,28,23,31)(25,32,26,29);(17,22,24,23)(18,25,27,26)(19,28,21,31)(20,29,30,32)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=f,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,c^b=ch,e^b=ei,g^c=gi,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, 6342865);