#10183106

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

4218 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gi,d^2=hi,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, 10090302);