#200634

order	512
nilpotency class	3
isoclinism	isoclinic class of #200400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁶	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1734 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,e^2=hi,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 107830);