#234861

order	512
nilpotency class	3
isoclinism	isoclinic class of #234851
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
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1380 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142057);