#10166303

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

4083 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=ghi,d^2=gh,e^2=ghi,f^2=gh,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, 10073499);