#439376

order	512
nilpotency class	3
isoclinism	isoclinic class of #417778
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

949 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=efg,c^2=f,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=dei,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, 346572);