#355006

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator	C₂³	C₂²×C₄²
socle, prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1186 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ef,d^2=eh,g^2=i,h^2=i,b^a=be,c^a=cf,d^c=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262202);