#6373699

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
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×#121	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1583 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,27,22,24,21,26)(19,25,29,20,28,30,32,31);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(18,24)(19,28)(26,27)(29,32);(10,14)(13,16)(17,19)(18,25)(20,27)(21,32)(22,28)(23,29)(24,30)(26,31);(17,20,23,30,22,31,21,25)(18,19,27,29,24,28,26,32);(1,3,4,7)(2,6,5,8)(17,21,22,23)(18,26,24,27)(19,29,28,32)(20,30,31,25)

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ci,e^c=ei,e^d=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280895);