#10166294

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

4066 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)(3,8)(4,6)(5,12)(7,11)(9,14)(10,15)(13,16)(17,18,23,24)(19,21,26,31)(20,25,29,32)(22,30,28,27)(33,34,39,40)(35,36,37,41)(38,46,47,42)(43,48,45,44);(1,3,6,11)(2,7,4,8)(5,10,14,13)(9,15,12,16)(17,19)(18,21)(20,28)(22,29)(23,26)(24,31)(25,27)(30,32)(34,40)(35,37)(42,46)(43,45);(2,4)(7,8)(9,12)(15,16)(18,24)(19,26)(20,29)(27,30)(33,35,39,37)(34,41,40,36)(38,43,47,45)(42,48,46,44);(1,4,6,2)(3,8,11,7)(5,12,14,9)(10,16,13,15)(17,20,23,29)(18,25,24,32)(19,28,26,22)(21,27,31,30)(33,36,39,41)(34,35,40,37)(38,44,47,48)(42,43,46,45);(1,3)(2,8)(4,7)(5,13)(6,11)(9,15)(10,14)(12,16)(17,21)(18,26)(19,24)(20,30)(22,32)(23,31)(25,28)(27,29)(33,37,39,35)(34,41,40,36)(38,45,47,43)(42,48,46,44);(1,5)(2,9)(3,10)(4,12)(6,14)(7,15)(8,16)(11,13)(17,22,23,28)(18,27,24,30)(19,29,26,20)(21,25,31,32)(33,38)(34,42)(35,43)(36,44)(37,45)(39,47)(40,46)(41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=ghi,d^2=h,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, 10073490);