#10068887

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

4135 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=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, 9976083);