#10247431

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

4101 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=i,f^2=h,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, 10154627);