#10181208

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

4068 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=gi,d^2=i,e^2=ghi,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, 10088404);