#10247562

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

3999 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)(4,8)(5,10)(6,7)(11,16)(12,13)(14,15)(17,18,24,29)(19,23,28,25)(20,26,21,30)(22,27,31,32);(1,3)(2,5)(4,12)(6,10)(7,9)(8,16)(11,14)(13,15)(17,19,24,28)(18,25,29,23)(20,22,21,31)(26,32,30,27)(33,34,38,41)(35,36,43,39)(37,40,46,48)(42,44,45,47);(2,7)(3,10)(8,15)(11,12)(17,20,24,21)(18,26,29,30)(19,22,28,31)(23,27,25,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,48)(45,46);(1,3)(2,5)(4,11)(6,10)(7,9)(8,13)(12,14)(15,16)(17,21,24,20)(18,26,29,30)(19,22,28,31)(23,32,25,27)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,48,45);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,16)(10,12)(17,22)(18,27)(19,21)(20,28)(23,30)(24,31)(25,26)(29,32)(33,35)(34,36)(37,45)(38,43)(39,41)(40,47)(42,46)(44,48);(1,5)(2,3)(4,13)(6,9)(7,10)(8,11)(12,15)(14,16)(17,23,24,25)(18,28,29,19)(20,27,21,32)(22,26,31,30)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,48)(43,45)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,d^2=gi,f^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, 10154758);