#149291

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₈	C₂³
commutator	C₂³	C₂²×C₁₆
socle, prime radical	C₂⁴	C₂²×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

618 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,g^2=h,h^2=i,b^a=bd,c^a=ce,c^b=cf

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56487);