#8103925

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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=i,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, 8011121);