#6391863

order	512
nilpotency class	2
isoclinism	isoclinic class of #46350
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×#525	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1647 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18,23,26,22,27,21,25)(19,28,24,32,29,20,30,31);(1,3)(2,5)(4,8)(6,7)(17,19,23,24,22,29,21,30)(18,20,26,31,27,28,25,32);(10,14)(15,16)(17,20)(18,24)(19,25)(21,32)(22,28)(23,31)(26,29)(27,30);(3,8)(5,7)(9,11,13,12)(10,15,14,16)(17,21,22,23)(18,25,27,26)(19,24,29,30)(20,31,28,32);(2,6)(5,7)(9,12,13,11)(10,16,14,15)(17,21,22,23)(18,26,27,25)(19,24,29,30)(20,32,28,31)

Polycyclic presentation:

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

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, 6299059);