#6412096

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	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1633 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)(9,10,14,16,13,11,12,15)(17,18)(19,26)(20,24)(21,30)(22,25)(23,28)(27,32)(29,31);(1,3)(2,5)(4,7)(6,8)(10,11)(15,16)(17,19)(18,23)(20,29)(21,27)(22,28)(24,32)(25,26)(30,31);(2,6)(3,7)(9,11,14,15,13,10,12,16);(2,6)(5,8)(9,12,13,14)(10,15,11,16)(17,20,22,21)(18,24,25,30)(19,27,28,29)(23,31,26,32);(9,12,13,14)(10,15,11,16)(17,21,22,20)(18,24,25,30)(19,27,28,29)(23,32,26,31)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cgi,d^b=dfi

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, 6319292);