#10247197

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

4020 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)(3,10)(4,13)(5,8)(6,7)(9,14)(11,15)(12,16)(17,18,24,29)(19,23,28,25)(20,26,21,30)(22,27,31,32);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,14)(13,16)(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);(1,4,7,14)(2,9,6,13)(3,11,5,12)(8,15,10,16)(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);(1,4)(2,9)(3,12)(5,11)(6,13)(7,14)(8,16)(10,15)(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);(1,5,7,3)(2,8,6,10)(4,12,14,11)(9,15,13,16)(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);(1,6,7,2)(3,10,5,8)(4,13,14,9)(11,16,12,15)(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=ghi,c^2=g,d^2=ghi,e^2=h,f^2=g,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, 10154393);