#9027286

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

4243 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=hi,e^2=h,f^2=i,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, 8934482);