#8622224

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

4168 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)(17,18,21,24)(19,22,26,25)(20,23,29,28)(27,30,32,31)(33,34,39,41)(35,40,45,48)(36,42,43,37)(38,44,47,46);(2,6)(3,10)(8,14)(11,12)(17,18)(19,25)(20,23)(21,24)(22,26)(27,31)(28,29)(30,32)(33,35,39,45)(34,40,41,48)(36,44,43,46)(37,38,42,47);(2,6)(7,9)(8,14)(15,16)(19,26)(22,25)(27,32)(30,31)(34,41)(35,45)(37,42)(44,46);(1,3,5,10)(2,7,6,9)(4,12,13,11)(8,16,14,15)(17,18)(19,22)(20,28)(21,24)(23,29)(25,26)(27,31)(30,32)(33,36,39,43)(34,42,41,37)(35,46,45,44)(38,48,47,40);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,27)(21,26)(23,30)(24,25)(28,31)(29,32)(33,37,39,42)(34,43,41,36)(35,47,45,38)(40,44,48,46);(1,4,5,13)(2,8,6,14)(3,11,10,12)(7,15,9,16)(17,20)(18,23)(19,27)(21,29)(22,30)(24,28)(25,31)(26,32)(33,38,39,47)(34,44,41,46)(35,42,45,37)(36,40,43,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=h,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, 8529420);