#251017

order	512
nilpotency class	3
isoclinism	isoclinic class of #250903
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1163 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:

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

Polycyclic presentation:

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

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, 158213);