#449876

order	512
nilpotency class	3
isoclinism	isoclinic class of #417692
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1068 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 357072);