#10183091

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

4157 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gi,d^2=hi,e^2=gi,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, 10090287);