#6435663

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1570 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)(3,6)(4,5)(7,8)(11,15)(14,16)(17,18,21,25,20,22,28,23)(19,24,27,31,26,29,32,30);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(18,22)(23,25)(24,29)(30,31);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(18,23)(21,28)(22,25)(24,30)(27,32)(29,31);(9,11,12,15)(10,14,13,16)(17,19)(18,24)(20,26)(21,27)(22,29)(23,30)(25,31)(28,32);(19,26)(24,29)(27,32)(30,31)

Polycyclic presentation:

2^9|a^2=g,c^2=fi,d^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,c^b=ch,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, 6342859);