#396981

order	512
nilpotency class	3
isoclinism	isoclinic class of #393909
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

1206 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=e,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 304177);