#8588370

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

4222 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)(3,9)(4,6)(5,8)(7,10)(11,16)(12,14)(13,15)(17,18)(19,20)(21,23)(22,24)(25,28)(26,27)(29,31)(30,32)(33,34,38,41)(35,42,37,39)(36,40,45,48)(43,47,44,46);(2,4)(3,10)(8,12)(11,13)(17,19,23,25)(18,20,21,28)(22,26,31,32)(24,27,29,30)(33,35,38,37)(34,39,41,42)(36,44,45,43)(40,47,48,46);(2,4)(7,9)(8,12)(15,16)(18,21)(19,25)(24,29)(26,32)(36,45)(40,48)(43,44)(46,47);(1,3)(2,7)(4,9)(5,13)(6,10)(8,16)(11,14)(12,15)(17,20)(18,19)(21,25)(22,30)(23,28)(24,32)(26,29)(27,31)(33,34)(35,39)(36,40)(37,42)(38,41)(43,46)(44,47)(45,48);(1,4,6,2)(3,9,10,7)(5,12,14,8)(11,16,13,15)(17,21,23,18)(19,20,25,28)(22,29,31,24)(26,27,32,30)(33,36)(34,40)(35,43)(37,44)(38,45)(39,46)(41,48)(42,47);(1,5,6,14)(2,8,4,12)(3,11,10,13)(7,15,9,16)(17,22)(18,24)(19,26)(20,27)(21,29)(23,31)(25,32)(28,30)(33,37,38,35)(34,39,41,42)(36,44,45,43)(40,46,48,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,e^2=h,f^2=g,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, 8495566);