#8733950

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

4239 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8641146);