#9173149

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

4330 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)(12,13,14,16)(17,18)(19,26)(20,24)(21,23)(22,25)(27,32)(28,29)(30,31)(33,34,38,41)(35,42,39,36)(37,40,47,48)(43,46,45,44);(1,3)(2,7)(4,12)(5,13)(6,11)(8,10)(9,14)(15,16)(17,19,22,21)(18,23,25,26)(20,27,28,30)(24,31,29,32)(33,35)(34,36)(37,45)(38,39)(40,46)(41,42)(43,47)(44,48);(3,11)(7,8)(12,14)(13,16)(20,28)(24,29)(27,30)(31,32)(33,36,38,42)(34,39,41,35)(37,44,47,46)(40,45,48,43);(1,3)(2,8)(4,14)(5,13)(6,11)(7,10)(9,12)(15,16)(17,20,22,28)(18,24,25,29)(19,27,21,30)(23,31,26,32)(33,36)(34,35)(37,46)(38,42)(39,41)(40,45)(43,48)(44,47);(1,4)(2,5)(3,12)(6,9)(7,13)(8,16)(10,15)(11,14)(17,18)(19,23)(20,29)(21,26)(22,25)(24,28)(27,32)(30,31)(33,37)(34,40)(35,43)(36,44)(38,47)(39,45)(41,48)(42,46);(1,5)(2,9)(3,13)(4,10)(6,15)(7,14)(8,12)(11,16)(17,21,22,19)(18,23,25,26)(20,30,28,27)(24,31,29,32)(33,35)(34,36)(37,45)(38,39)(40,46)(41,42)(43,47)(44,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=gh,f^2=gh,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, 9080345);