#8104726

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

4930 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,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, 8011922);