#6422184

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
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	C₂×#547	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1727 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=f,d^2=h,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=efi,c^b=cg,d^b=df

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6329380);