#6407433

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, prime radical	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1330 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=g,d^2=f,e^2=fgh,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ci,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, 6314629);