#450118

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	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1101 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:

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

Polycyclic presentation:

2^9|a^2=ehi,b^2=ef,d^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, 357314);