#9173145

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

4140 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)(17,18,22,20)(19,25,27,23)(21,24,30,28)(26,32,29,31)(33,34)(35,41)(36,47)(37,40)(38,42)(39,44)(43,46)(45,48);(1,3)(2,7)(4,12)(5,13)(6,11)(8,10)(9,14)(15,16)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,30)(28,31)(33,35,40,48)(34,41,37,45)(36,43,42,44)(38,39,47,46);(3,11)(7,8)(12,14)(13,16)(18,20)(23,25)(24,28)(31,32)(33,36,40,42)(34,38,37,47)(35,44,48,43)(39,41,46,45);(1,3)(2,8)(4,14)(5,13)(6,11)(7,10)(9,12)(15,16)(17,20,22,18)(19,25,27,23)(21,24,30,28)(26,31,29,32)(33,37,40,34)(35,45,48,41)(36,38,42,47)(39,44,46,43);(1,4)(2,5)(3,12)(6,9)(7,13)(8,16)(10,15)(11,14)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)(33,38)(34,42)(35,46)(36,37)(39,48)(40,47)(41,43)(44,45);(1,5)(2,9)(3,13)(4,10)(6,15)(7,14)(8,12)(11,16)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,30)(28,31)(33,39,40,46)(34,43,37,44)(35,47,48,38)(36,41,42,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=h,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, 9080341);