#9027831

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

4210 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(33,34,38,41)(35,39,43,37)(36,40,46,48)(42,47,44,45);(2,6)(7,9)(8,12)(14,15)(17,19,23,20)(18,24,22,25)(21,28,29,27)(26,32,30,31)(33,35)(34,39)(36,42)(37,41)(38,43)(40,47)(44,46)(45,48);(2,6)(3,10)(4,13)(14,15)(18,22)(19,20)(21,29)(31,32)(33,35,38,43)(34,39,41,37)(36,44,46,42)(40,45,48,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,20,23,19)(18,25,22,24)(21,27,29,28)(26,31,30,32)(33,35)(34,39)(36,44)(37,41)(38,43)(40,45)(42,46)(47,48);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,21,23,29)(18,26,22,30)(19,27,20,28)(24,31,25,32)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,48)(43,44);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,22,23,18)(19,25,20,24)(21,30,29,26)(27,32,28,31)(33,37,38,39)(34,35,41,43)(36,45,46,47)(40,42,48,44)

Polycyclic presentation:

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

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, 8935027);