#10247670

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

3986 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)(3,7)(4,9)(5,13)(6,8)(10,15)(11,12)(14,16)(17,18,21,24)(19,25,27,22)(20,23,30,29)(26,32,28,31)(33,34,40,45)(35,39,44,41)(36,42,37,46)(38,43,47,48);(1,3,6,9)(2,7,8,4)(5,11,15,16)(10,14,13,12)(17,19,21,27)(18,22,24,25)(20,28,30,26)(23,32,29,31)(33,35,40,44)(34,41,45,39)(36,38,37,47)(42,48,46,43);(2,8)(3,9)(10,13)(11,16)(17,19,21,27)(18,22,24,25)(20,26,30,28)(23,31,29,32)(33,36,40,37)(34,42,45,46)(35,38,44,47)(39,43,41,48);(1,2)(3,4)(5,10)(6,8)(7,9)(11,12)(13,15)(14,16)(17,19)(18,22)(20,28)(21,27)(23,32)(24,25)(26,30)(29,31)(33,37,40,36)(34,42,45,46)(35,38,44,47)(39,48,41,43);(1,4)(2,9)(3,8)(5,14)(6,7)(10,11)(12,15)(13,16)(17,20)(18,23)(19,26)(21,30)(22,31)(24,29)(25,32)(27,28)(33,38)(34,43)(35,37)(36,44)(39,46)(40,47)(41,42)(45,48);(1,5,6,15)(2,10,8,13)(3,11,9,16)(4,12,7,14)(17,18)(19,22)(20,29)(21,24)(23,30)(25,27)(26,32)(28,31)(33,39,40,41)(34,44,45,35)(36,43,37,48)(38,42,47,46)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=ghi,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, 10154866);