#6405648

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	#52129	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1716 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,e^2=fh,h^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ci,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, 6312844);