#9184698

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

4250 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)(4,7)(5,9)(6,12)(10,15)(11,13)(14,16)(17,18)(19,26)(20,24)(21,23)(22,25)(27,32)(28,29)(30,31)(33,34,38,39)(35,41,46,40)(36,42,43,37)(44,45,47,48);(1,3,7,9)(2,8,4,5)(6,11,15,16)(10,14,12,13)(17,19,22,21)(18,23,25,26)(20,27,28,30)(24,31,29,32)(34,39)(36,43)(40,41)(44,47);(2,4)(3,9)(10,12)(11,16)(20,28)(24,29)(27,30)(31,32)(33,35,38,46)(34,40,39,41)(36,44,43,47)(37,45,42,48);(1,4,7,2)(3,5,9,8)(6,12,15,10)(11,13,16,14)(17,20,22,28)(18,24,25,29)(19,27,21,30)(23,31,26,32)(33,35)(34,41)(36,47)(37,45)(38,46)(39,40)(42,48)(43,44);(1,5)(2,9)(3,4)(6,14)(7,8)(10,11)(12,16)(13,15)(17,18)(19,23)(20,29)(21,26)(22,25)(24,28)(27,32)(30,31)(33,36)(34,42)(35,44)(37,39)(38,43)(40,48)(41,45)(46,47);(1,6)(2,10)(3,11)(4,12)(5,13)(7,15)(8,14)(9,16)(17,21,22,19)(18,23,25,26)(20,30,28,27)(24,31,29,32)(33,37,38,42)(34,43,39,36)(35,45,46,48)(40,47,41,44)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=hi,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=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, 9091894);