#450736

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

756 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)(9,10,13,15)(11,14,12,16)(17,18,22,28,21,27,23,29)(19,20,25,32,26,24,30,31);(1,3,5,8)(2,6,4,7)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,27)(21,26)(22,25)(23,30)(28,31)(29,32);(1,2,5,4)(3,6,8,7)(9,12,13,11)(10,14,15,16)(17,20,21,24)(18,25,27,30)(19,28,26,29)(22,32,23,31);(1,4,5,2)(3,6,8,7)(17,20,21,24)(18,26,27,19)(22,31,23,32)(25,29,30,28)

Polycyclic presentation:

2^9|a^2=efh,b^2=efi,c^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, 357932);