#9160746

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) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9067942);