#232974

order	512
nilpotency class	3
isoclinism	isoclinic class of #232963
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1220 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)(3,10)(4,15)(5,8)(6,9)(7,13)(11,14)(12,16)(17,18)(19,23)(20,25)(21,30)(22,26)(24,27)(28,32)(29,31);(1,3)(2,7)(4,12)(5,13)(6,14)(8,11)(9,16)(10,15)(17,19)(18,23)(20,29)(21,27)(22,28)(24,30)(25,31)(26,32);(1,2)(3,11)(4,8)(5,15)(6,9)(7,13)(10,14)(12,16)(17,19)(18,24)(20,27)(21,29)(22,28)(23,30)(25,32)(26,31);(1,4,6,5)(2,8,9,15)(3,12,14,13)(7,11,16,10)(17,20,22,21)(18,25,26,30)(19,27,28,29)(23,24,32,31)

Polycyclic presentation:

2^9|d^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,c^b=cg,d^b=di,e^b=eh,g^b=gh,f^c=fi,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, 140170);