#9210323

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

4092 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)(4,11)(5,8)(6,9)(10,15)(12,16)(13,14)(17,18)(19,25)(20,23)(21,24)(22,27)(26,32)(28,31)(29,30)(33,34)(35,44)(36,41)(37,42)(38,40)(39,43)(45,46)(47,48);(1,3,6,4)(2,7,9,11)(5,10,14,12)(8,15,13,16)(17,19)(18,22)(20,28)(21,27)(23,32)(24,25)(26,30)(29,31)(33,35,39,38)(34,40,43,44)(36,45,42,47)(37,46,41,48);(3,4)(7,11)(10,12)(15,16)(17,18,21,24)(19,22,27,25)(20,23,30,29)(26,31,28,32)(33,36,39,42)(34,41,43,37)(35,45,38,47)(40,48,44,46);(1,4,6,3)(2,7,9,11)(5,12,14,10)(8,15,13,16)(17,18)(19,22)(20,29)(21,24)(23,30)(25,27)(26,32)(28,31)(33,36)(34,41)(35,45)(37,43)(38,47)(39,42)(40,48)(44,46);(1,2)(3,7)(4,11)(5,13)(6,9)(8,14)(10,16)(12,15)(17,20,21,30)(18,23,24,29)(19,26,27,28)(22,31,25,32)(33,37,39,41)(34,42,43,36)(35,46,38,48)(40,47,44,45);(1,5,6,14)(2,8,9,13)(3,10,4,12)(7,15,11,16)(17,19)(18,22)(20,28)(21,27)(23,32)(24,25)(26,30)(29,31)(33,38,39,35)(34,40,43,44)(36,47,42,45)(37,46,41,48)

Polycyclic presentation:

2^9|b^2=ghi,c^2=g,d^2=i,e^2=g,f^2=ghi,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, 9117519);