#235521

order	512
nilpotency class	3
isoclinism	isoclinic class of #235516
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	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1257 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=ghi,c^b=cg,d^b=di,e^b=eh,f^b=fh,d^c=dhi,e^c=ei,f^c=fhi

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, 142717);