#10158436

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

4234 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=i,e^2=gi,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, 10065632);