#6424556

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	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1391 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)(3,7)(4,6)(5,8)(10,14)(11,12)(17,18,23,26,22,24,21,20)(19,25,28,32,29,31,30,27);(3,8)(5,7)(9,10)(11,16)(12,15)(13,14)(18,24)(20,26)(25,31)(27,32);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,15,14,16)(17,19)(18,25)(20,27)(21,30)(22,29)(23,28)(24,31)(26,32);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,20,21,24,22,26,23,18)(19,27,30,31,29,32,28,25);(9,12,13,11)(10,15,14,16)(17,21,22,23)(18,26,24,20)(19,28,29,30)(25,27,31,32)

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=fh,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df,e^c=ei,e^d=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, 6331752);