#9382272

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

4182 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(18,20)(19,25)(24,28)(26,31)(33,34,38,36)(35,39,43,41)(37,44,45,40)(42,47,48,46);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,18,22,20)(19,23,25,27)(21,28,29,24)(26,32,31,30)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(2,3)(7,9)(8,10)(14,15)(19,25)(23,27)(26,31)(30,32)(34,36)(39,41)(40,44)(46,47);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,36,38,34)(35,41,43,39)(37,44,45,40)(42,47,48,46);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,20,22,18)(19,23,25,27)(21,28,29,24)(26,30,31,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,21,22,29)(18,24,20,28)(19,26,25,31)(23,30,27,32)(33,37,38,45)(34,40,36,44)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dghi,e^d=egi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9289468);