#9175005

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

4185 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)(3,10)(4,8)(5,7)(6,9)(11,16)(12,15)(13,14)(17,18)(19,23)(20,26)(21,24)(22,25)(27,31)(28,32)(29,30)(33,34)(35,41)(36,39)(37,40)(38,43)(42,48)(44,47)(45,46);(1,3,6,5)(2,7,9,10)(4,11,12,13)(8,14,15,16)(17,19,22,26)(18,23,25,20)(21,27,30,32)(24,31,29,28)(33,35)(34,38)(36,44)(37,43)(39,48)(40,41)(42,46)(45,47);(4,12)(8,15)(11,13)(14,16)(19,26)(20,23)(27,32)(28,31)(33,34,37,40)(35,38,43,41)(36,39,46,45)(42,47,44,48);(1,4,6,12)(2,8,9,15)(3,11,5,13)(7,14,10,16)(17,20)(18,19)(21,28)(22,23)(24,27)(25,26)(29,32)(30,31)(33,34)(35,38)(36,45)(37,40)(39,46)(41,43)(42,48)(44,47);(1,5,6,3)(2,7,9,10)(4,11,12,13)(8,16,15,14)(17,18)(19,23)(20,26)(21,29)(22,25)(24,30)(27,28)(31,32)(33,36,37,46)(34,39,40,45)(35,42,43,44)(38,47,41,48);(1,2)(3,7)(4,8)(5,10)(6,9)(11,14)(12,15)(13,16)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,35)(34,38)(36,44)(37,43)(39,48)(40,41)(42,46)(45,47)

Polycyclic presentation:

2^9|b^2=ghi,c^2=h,d^2=gh,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, 9082201);