#6408037

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
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×D₄	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1363 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)(9,10,14,15,13,16,11,12)(17,18)(19,27)(20,24)(21,26)(22,25)(23,29)(28,32)(30,31);(1,3)(2,5)(4,7)(6,8)(9,11,13,14)(10,15,16,12)(17,19,22,29)(18,23,25,27)(20,30,21,28)(24,32,26,31);(2,6)(3,7)(9,11,13,14)(10,12,16,15)(17,20,22,21)(18,24,25,26)(19,28,29,30)(23,31,27,32);(2,6)(5,8)(18,25)(19,29)(24,26)(28,30);(9,12,11,16,13,15,14,10)(17,21,22,20)(18,26,25,24)(19,28,29,30)(23,31,27,32)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=cfgi,d^b=dfi,e^b=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, 6315233);