#9173158

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

4158 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)(17,18,23,22)(19,26,21,24)(20,25,28,29)(27,32,30,31)(33,34)(35,37)(36,45)(38,41)(39,42)(40,44)(43,46)(47,48);(1,3)(2,7)(4,12)(5,13)(6,11)(8,10)(9,14)(15,16)(17,19,23,21)(18,24,22,26)(20,27,28,30)(25,31,29,32)(33,35)(34,40)(36,43)(37,42)(38,48)(39,44)(41,46)(45,47);(3,11)(7,8)(12,14)(13,16)(20,28)(25,29)(27,30)(31,32)(33,36,39,38)(34,41,42,45)(35,43,44,48)(37,47,40,46);(1,3)(2,8)(4,14)(5,13)(6,11)(7,10)(9,12)(15,16)(17,20)(18,25)(19,27)(21,30)(22,29)(23,28)(24,31)(26,32)(33,37,39,40)(34,35,42,44)(36,46,38,47)(41,48,45,43);(1,4)(2,5)(3,12)(6,9)(7,13)(8,16)(10,15)(11,14)(17,21,23,19)(18,24,22,26)(20,27,28,30)(25,32,29,31)(33,38,39,36)(34,41,42,45)(35,43,44,48)(37,46,40,47);(1,5)(2,9)(3,13)(4,10)(6,15)(7,14)(8,12)(11,16)(17,22,23,18)(19,26,21,24)(20,29,28,25)(27,32,30,31)(33,35)(34,40)(36,43)(37,42)(38,48)(39,44)(41,46)(45,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=h,e^2=g,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, 9080354);