#8104420

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5060 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,d^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8011616);