#6402144

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
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	#2703	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1401 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)(9,10,13,16)(11,15,12,14)(17,18,23,21,22,26,27,32)(19,24,20,29,28,31,30,25);(1,3,4,7)(2,5,6,8)(9,10)(11,14)(12,15)(13,16)(17,19,22,28)(18,24,26,31)(20,27,30,23)(21,29,32,25);(2,6)(5,8)(9,11,13,12)(10,14,16,15)(17,20)(18,25)(19,27)(21,24)(22,30)(23,28)(26,29)(31,32);(9,11,13,12)(10,15,16,14)(19,28)(20,30)(24,31)(25,29);(9,12,13,11)(10,15,16,14)(17,21,23,26,22,32,27,18)(19,29,20,31,28,25,30,24)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=f,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,e^a=efi,d^b=dfi,d^c=di,e^c=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, 6309340);