#9167720

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

4148 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)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(19,26)(21,29)(23,24)(25,28)(33,34,39,44)(35,42,37,40)(36,46,47,41)(38,43,45,48);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,18)(19,24)(20,22)(21,25)(23,26)(27,31)(28,29)(30,32)(33,35,39,37)(34,40,44,42)(36,38,47,45)(41,43,46,48);(3,10)(6,7)(11,16)(14,15)(17,19,22,26)(18,23,20,24)(21,27,29,32)(25,30,28,31)(33,36,39,47)(34,41,44,46)(35,45,37,38)(40,48,42,43);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,19)(18,24)(20,23)(21,27)(22,26)(25,31)(28,30)(29,32)(33,37,39,35)(34,42,44,40)(36,38,47,45)(41,43,46,48);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,20,22,18)(19,24,26,23)(21,25,29,28)(27,30,32,31)(33,38)(34,43)(35,36)(37,47)(39,45)(40,41)(42,46)(44,48);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,48)(46,47)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9074916);