#465969

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
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	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

926 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,18)(19,20)(21,26)(22,27)(23,28)(24,30)(25,31)(29,32);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19,23,29,21,24,22,25)(18,20,28,32,26,30,27,31);(1,3)(2,6)(4,7)(5,8)(10,15)(14,16)(17,20)(18,24)(19,26)(21,30)(22,32)(23,31)(25,27)(28,29);(3,7)(6,8)(9,12,13,11)(10,16,15,14)(17,20)(18,25)(19,28)(21,30)(22,31)(23,32)(24,27)(26,29)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 373165);