#10247202

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

3989 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:

(2,7)(3,10)(5,13)(12,15)(17,18,23,28)(19,25,21,24)(20,30,31,26)(22,27,32,29);(1,2,6,7)(3,4,10,8)(5,9,13,14)(11,12,16,15)(17,19,23,21)(18,24,28,25)(20,27,31,29)(22,26,32,30)(33,34,38,41)(35,36,43,39)(37,40,46,48)(42,44,45,47);(1,3,6,10)(2,8,7,4)(5,11,13,16)(9,15,14,12)(17,19,23,21)(18,25,28,24)(20,29,31,27)(22,26,32,30)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,48)(45,46);(1,3)(2,4)(5,11)(6,10)(7,8)(9,12)(13,16)(14,15)(17,20)(18,26)(19,29)(21,27)(22,24)(23,31)(25,32)(28,30)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,48,45);(1,4)(2,3)(5,12)(6,8)(7,10)(9,11)(13,15)(14,16)(17,21,23,19)(18,24,28,25)(20,29,31,27)(22,26,32,30)(33,35)(34,36)(37,45)(38,43)(39,41)(40,47)(42,46)(44,48);(1,5,6,13)(2,9,7,14)(3,11,10,16)(4,12,8,15)(17,22,23,32)(18,27,28,29)(19,26,21,30)(20,24,31,25)(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=hi,e^2=gh,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, 10154398);