#450828

order	512
nilpotency class	3
isoclinism	isoclinic class of #417692
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1108 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=e,d^2=ef,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, 358024);