#9390562

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

4364 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=hi,d^2=g,e^2=g,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, 9297758);