#9174996

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

4274 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)(3,7)(4,10)(5,8)(6,9)(11,15)(12,16)(13,14)(17,18,22,21)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,34,39,43)(35,37,42,40)(36,45,38,41)(44,46,48,47);(1,3,6,10)(2,7,9,4)(5,11,14,16)(8,15,13,12)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,32)(29,31)(33,35,39,42)(34,40,43,37)(36,44,38,48)(41,46,45,47);(3,10)(4,7)(11,16)(12,15)(20,28)(24,29)(26,32)(30,31)(33,36,39,38)(34,41,43,45)(35,44,42,48)(37,47,40,46);(1,4)(2,3)(5,12)(6,7)(8,11)(9,10)(13,16)(14,15)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)(33,37,39,40)(34,42,43,35)(36,46,38,47)(41,44,45,48);(1,2)(3,7)(4,10)(5,13)(6,9)(8,14)(11,12)(15,16)(17,21,22,18)(19,25,27,23)(20,24,28,29)(26,30,32,31)(33,38,39,36)(34,41,43,45)(35,44,42,48)(37,46,40,47);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,19)(18,25)(20,26)(21,23)(22,27)(24,31)(28,32)(29,30)(33,34)(35,40)(36,41)(37,42)(38,45)(39,43)(44,46)(47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=h,e^2=g,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, 9082192);