#6403034

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	#2699	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1407 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)(11,14)(12,16)(13,15)(17,18,23,28,22,27,21,26)(19,29,30,32,31,24,20,25);(1,3,4,7)(2,5,6,8)(9,11,14,10)(12,16,13,15)(17,19)(18,24)(20,21)(22,31)(23,30)(25,28)(26,32)(27,29);(2,6)(5,8)(9,12,14,13)(10,15,11,16)(17,20,22,30)(18,25,27,32)(19,21,31,23)(24,28,29,26);(9,12,14,13)(10,16,11,15)(17,21,22,23)(18,26,27,28)(19,30,31,20)(24,25,29,32);(9,13,14,12)(10,16,11,15)(17,18,21,26,22,27,23,28)(19,29,20,25,31,24,30,32)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=f,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,e^a=ef,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, 6310230);