#8103920

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

4166 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,11)(4,5)(6,9)(7,10)(8,14)(12,15)(13,16)(17,18)(19,22)(20,25)(21,26)(23,28)(24,30)(27,31)(29,32)(33,34,37,40)(35,38,42,46)(36,39,44,48)(41,45,43,47);(1,3,7,14)(2,8,10,11)(4,12,9,16)(5,13,6,15)(17,19,22,18)(20,25,28,23)(21,24,30,26)(27,31,29,32)(33,35,37,42)(34,38,40,46)(36,43,44,41)(39,47,48,45);(1,4,7,9)(2,5,10,6)(3,12,14,16)(8,13,11,15)(17,20,22,28)(18,23,19,25)(21,29,30,27)(24,32,26,31)(33,36,37,44)(34,39,40,48)(35,41,42,43)(38,45,46,47);(1,5,7,6)(2,9,10,4)(3,13,14,15)(8,16,11,12)(17,21,22,30)(18,24,19,26)(20,27,28,29)(23,31,25,32)(36,44)(39,48)(41,43)(45,47);(3,14)(8,11)(12,16)(13,15)(18,19)(23,25)(24,26)(31,32)(35,42)(38,46)(41,43)(45,47);(1,6,7,5)(2,9,10,4)(3,15,14,13)(8,16,11,12)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gi,d^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, 8011116);