#8105345

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

4150 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)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18)(19,25)(20,28)(21,24)(22,27)(23,29)(26,30)(31,32)(33,34,38,41)(35,43,45,39)(36,42,37,40)(44,47,46,48);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32)(33,35)(34,39)(36,46)(37,44)(38,45)(40,48)(41,43)(42,47);(1,2)(3,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,20)(18,23)(19,26)(21,29)(22,30)(24,28)(25,31)(27,32)(33,36,38,37)(34,40,41,42)(35,44,45,46)(39,47,43,48);(2,7)(6,9)(8,14)(13,15)(18,24)(19,27)(20,29)(30,31)(34,41)(35,45)(40,42)(44,46);(3,10)(6,9)(11,16)(13,15)(18,24)(19,27)(23,28)(26,32)(33,37,38,36)(34,42,41,40)(35,44,45,46)(39,47,43,48);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)

Polycyclic presentation:

2^9|a^2=i,c^2=i,e^2=i,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, 8012541);