#473440

order	512
nilpotency class	3
isoclinism	isoclinic class of #454382
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1066 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fgh,d^2=efi,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de,g^c=gi

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, 380636);