#405470

order	512
nilpotency class	3
isoclinism	isoclinic class of #393940
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1746 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)(3,4)(6,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,26)(19,21,25,22)(20,28,32,27)(24,30,29,31);(5,6,8,7)(9,11,12,15)(10,13,14,16)(17,19)(18,21)(20,29)(22,26)(23,25)(24,32)(27,30)(28,31);(2,4)(5,7,8,6)(17,20)(18,24)(19,27)(21,30)(22,31)(23,32)(25,28)(26,29);(10,14)(13,16)(17,21,23,22)(18,25,26,19)(20,30,32,31)(24,28,29,27)

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=e,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312666);