#8622247

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

4113 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)(4,6)(5,8)(7,10)(11,16)(12,14)(13,15)(17,18)(19,25)(20,22)(21,24)(23,27)(26,32)(28,30)(29,31)(33,34,39,41)(35,40,45,48)(36,42,43,37)(38,44,47,46);(2,4)(3,10)(8,12)(11,13)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,30)(28,31)(33,35,39,45)(34,40,41,48)(36,44,43,46)(37,38,42,47);(2,4)(7,9)(8,12)(15,16)(21,30)(24,28)(26,29)(31,32)(34,41)(35,45)(37,42)(44,46);(1,3)(2,7)(4,9)(5,13)(6,10)(8,16)(11,14)(12,15)(17,20,22,18)(19,25,27,23)(21,28,30,24)(26,32,29,31)(33,36,39,43)(34,42,41,37)(35,46,45,44)(38,48,47,40);(1,4,6,2)(3,9,10,7)(5,12,14,8)(11,16,13,15)(17,21,22,30)(18,24,20,28)(19,26,27,29)(23,31,25,32)(33,37,39,42)(34,43,41,36)(35,47,45,38)(40,44,48,46);(1,5,6,14)(2,8,4,12)(3,11,10,13)(7,15,9,16)(17,19)(18,25)(20,23)(21,26)(22,27)(24,32)(28,31)(29,30)(33,38,39,47)(34,44,41,46)(35,42,45,37)(36,40,43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=gh,e^2=gi,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, 8529443);