#6409238

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#34935	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1937 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,d^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dfi,e^c=ei,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6316434);