#9175001

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

4299 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)(3,7)(4,10)(5,8)(6,9)(11,15)(12,16)(13,14)(17,18)(19,20)(21,25)(22,24)(23,27)(26,28)(29,30)(31,32)(33,34)(35,44)(36,40)(37,41)(38,42)(39,43)(45,46)(47,48);(1,3,6,10)(2,7,9,4)(5,11,14,16)(8,15,13,12)(17,19)(18,23)(20,24)(21,29)(22,27)(25,28)(26,31)(30,32)(33,35,39,42)(34,38,43,44)(36,45,41,48)(37,46,40,47);(3,10)(4,7)(11,16)(12,15)(18,24)(20,23)(25,32)(28,30)(33,36,39,41)(34,40,43,37)(35,45,42,48)(38,47,44,46);(1,4)(2,3)(5,12)(6,7)(8,11)(9,10)(13,16)(14,15)(17,20,22,23)(18,19,24,27)(21,30,31,28)(25,29,32,26)(33,36)(34,40)(35,45)(37,43)(38,47)(39,41)(42,48)(44,46);(1,2)(3,7)(4,10)(5,13)(6,9)(8,14)(11,12)(15,16)(17,21,22,31)(18,25,24,32)(19,26,27,29)(20,28,23,30)(33,37,39,40)(34,41,43,36)(35,46,42,47)(38,48,44,45);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,19)(18,23)(20,24)(21,29)(22,27)(25,28)(26,31)(30,32)(33,38)(34,42)(35,43)(36,47)(37,45)(39,44)(40,48)(41,46)

Polycyclic presentation:

2^9|b^2=ghi,c^2=gh,d^2=h,e^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, 9082197);