#8100328

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

4185 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,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, 8007524);