#10106311

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

4097 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gi,d^2=g,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, 10013507);