#396982

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

1184 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)(17,18,21,25)(19,22,27,31)(20,23,28,24)(26,29,32,30);(1,2)(3,4)(5,6,8,7)(10,14)(15,16);(2,4)(5,6,8,7)(9,11,13,12)(10,15,14,16)(17,19,21,27)(18,22,25,31)(20,26,28,32)(23,29,24,30);(5,7,8,6)(9,12,13,11)(10,16,14,15)(18,23)(20,28)(22,29)(24,25)(26,32)(30,31)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=ei,d^2=e,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, 304178);