#6406524

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
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	#2676	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1562 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)(11,16)(12,14)(13,15)(17,18,23,27,22,21,26,32)(19,28,30,25,29,24,20,31);(2,5)(3,7)(17,19)(18,24)(20,26)(21,28)(22,29)(23,30)(25,32)(27,31);(9,11,13,12)(10,14,15,16)(17,20,22,30)(18,25,21,31)(19,26,29,23)(24,32,28,27);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,14,15,16)(19,29)(20,30)(24,28)(25,31);(2,5)(6,8)(17,21,23,32,22,18,26,27)(19,24,30,31,29,28,20,25)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6313720);