#8104727

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

5083 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=gi,e^2=g,f^2=gh,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, 8011923);