#9243316

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

4203 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9150512);