#8622017

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4193 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8529213);