#10247332

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

4060 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10154528);