#8873807

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

4016 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

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, 8781003);