#8873722

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

4153 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)(3,9,10,8)(4,13,15,5)(11,16,14,12)(17,18,22,19)(20,25,28,23)(21,26,29,24)(27,30,32,31)(33,34,38,39)(35,42,40,36)(37,45,46,41)(43,44,47,48);(2,7)(5,13)(8,9)(12,16)(17,18,22,19)(20,25,28,23)(21,26,29,24)(27,30,32,31)(34,39)(36,42)(37,46)(43,47);(3,10)(8,9)(11,14)(12,16)(18,19)(23,25)(24,26)(30,31)(33,35,38,40)(34,36,39,42)(37,43,46,47)(41,44,45,48);(1,3)(2,8)(4,14)(5,12)(6,10)(7,9)(11,15)(13,16)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,30,32,31)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,48)(46,47);(1,4)(2,5)(3,11)(6,15)(7,13)(8,16)(9,12)(10,14)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,36)(34,40)(35,39)(37,44)(38,42)(41,47)(43,45)(46,48);(1,5)(2,4)(3,12)(6,13)(7,15)(8,14)(9,11)(10,16)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,37)(34,41)(35,43)(36,44)(38,46)(39,45)(40,47)(42,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=hi,d^2=i,e^2=i,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, 8780918);