#252907

order	512
nilpotency class	3
isoclinism	isoclinic class of #247527
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1054 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=e,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,f^c=fh,g^c=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 160103);