#8533191

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

4286 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=gi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8440387);