#10042513

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

4273 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)(3,8)(4,9)(5,13)(6,7)(10,14)(11,15)(12,16)(17,18)(19,22)(20,23)(21,26)(24,30)(25,27)(28,32)(29,31)(33,34)(35,38)(36,43)(37,39)(40,44)(41,47)(42,45)(46,48);(2,7)(3,10)(4,13)(12,15)(17,18)(19,22)(20,25)(21,24)(23,27)(26,30)(28,32)(29,31)(33,35,37,42)(34,38,39,45)(36,41,44,48)(40,46,43,47);(2,7)(3,10)(4,13)(12,15)(18,19)(20,27)(24,26)(28,29)(34,39)(38,45)(40,43)(46,47);(1,3,6,10)(2,8,7,14)(4,11,13,16)(5,12,9,15)(17,19,22,18)(20,25,27,23)(21,26,30,24)(28,32,29,31)(33,34)(35,38)(36,40)(37,39)(41,46)(42,45)(43,44)(47,48);(1,4,6,13)(2,9,7,5)(3,11,10,16)(8,15,14,12)(17,20)(18,23)(19,25)(21,29)(22,27)(24,32)(26,31)(28,30)(33,35)(34,38)(36,41)(37,42)(39,45)(40,46)(43,47)(44,48);(1,5,6,9)(2,4,7,13)(3,12,10,15)(8,11,14,16)(17,21,22,30)(18,24,19,26)(20,28,27,29)(23,31,25,32)(33,36,37,44)(34,40,39,43)(35,41,42,48)(38,46,45,47)

Polycyclic presentation:

2^9|b^2=ghi,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9949709);