#8602491

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
prime radical	#48409	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4051 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)(21,29)(22,26)(23,24)(25,32)(27,30)(28,31)(33,34,38,41)(35,39,42,46)(36,37,40,45)(43,44,47,48);(2,4)(3,10)(8,12)(11,13)(17,19,23,27)(18,20,24,30)(21,31,32,22)(25,26,29,28)(35,42)(39,46)(43,47)(44,48);(2,4)(7,9)(8,12)(15,16)(18,24)(19,27)(21,32)(26,28)(33,35,38,42)(34,39,41,46)(36,43,40,47)(37,44,45,48);(1,3)(2,7)(4,9)(5,13)(6,10)(8,16)(11,14)(12,15)(17,20)(18,19)(21,28)(22,25)(23,30)(24,27)(26,32)(29,31)(33,36)(34,37)(35,43)(38,40)(39,44)(41,45)(42,47)(46,48);(1,4,6,2)(3,9,10,7)(5,12,14,8)(11,16,13,15)(17,21)(18,25)(19,22)(20,28)(23,32)(24,29)(26,30)(27,31)(33,35)(34,39)(36,43)(37,44)(38,42)(40,47)(41,46)(45,48);(1,5,6,14)(2,8,4,12)(3,11,10,13)(7,15,9,16)(17,22)(18,26)(19,21)(20,29)(23,31)(24,28)(25,30)(27,32)(33,37)(34,40)(35,44)(36,41)(38,45)(39,47)(42,48)(43,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=ghi,e^2=g,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, 8509687);