#10106475

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

4240 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10013671);