#8633492

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

4168 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)(3,8)(4,10)(5,14)(6,12)(7,9)(11,13)(15,16)(33,34,38,37)(35,41,43,39)(36,44,46,40)(42,47,45,48);(1,3)(2,8)(4,13)(5,15)(6,9)(7,12)(10,11)(14,16)(17,18)(19,24)(20,21)(22,23)(25,29)(26,30)(27,32)(28,31)(33,35,38,43)(34,39,37,41)(36,45,46,42)(40,48,44,47);(2,9)(3,12)(5,13)(10,16)(18,23)(20,29)(24,31)(26,32)(33,35,38,43)(34,39,37,41)(36,42,46,45)(40,47,44,48);(1,4)(2,10)(3,5)(6,11)(7,15)(8,14)(9,16)(12,13)(17,19)(18,24)(20,26)(21,30)(22,28)(23,31)(25,27)(29,32)(33,36)(34,40)(35,42)(37,44)(38,46)(39,47)(41,48)(43,45);(1,5)(2,11)(3,4)(6,10)(7,13)(8,16)(9,14)(12,15)(17,20)(18,25)(19,26)(21,23)(22,29)(24,27)(28,32)(30,31)(33,35)(34,39)(36,42)(37,41)(38,43)(40,47)(44,48)(45,46);(1,6)(2,12)(3,9)(4,14)(5,10)(7,8)(11,15)(13,16)(17,21,22,25)(18,20,23,29)(19,27,28,30)(24,32,31,26)(33,37,38,34)(35,41,43,39)(36,40,46,44)(42,47,45,48)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8540688);