#232456

order	512
nilpotency class	3
isoclinism	isoclinic class of #232451
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1422 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gi,c^b=cg,d^b=dh,e^b=eh,g^b=gh,e^c=ei,f^c=fh,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139652);