#9160753

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

4145 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)(3,8)(4,12)(5,9)(6,10)(7,13)(11,15)(14,16)(17,18)(19,24)(20,26)(21,27)(22,30)(23,25)(28,31)(29,32)(33,34,38,41)(35,42,37,39)(36,44,46,40)(43,47,45,48);(1,3)(2,5)(4,11)(6,9)(7,15)(8,10)(12,16)(13,14)(17,19,23,28)(18,24,25,31)(20,29,30,21)(22,27,26,32)(33,34)(35,39)(36,40)(37,42)(38,41)(43,47)(44,46)(45,48);(4,13)(7,12)(11,14)(15,16)(18,25)(19,28)(20,30)(27,32)(33,35,38,37)(34,39,41,42)(36,45,46,43)(40,48,44,47);(1,4)(2,7)(3,11)(5,15)(6,13)(8,16)(9,14)(10,12)(17,20)(18,26)(19,29)(21,28)(22,25)(23,30)(24,32)(27,31)(33,36,38,46)(34,40,41,44)(35,43,37,45)(39,47,42,48);(1,3)(2,8)(4,14)(5,10)(6,9)(7,15)(11,13)(12,16)(17,21,23,29)(18,27,25,32)(19,30,28,20)(22,31,26,24)(33,37,38,35)(34,39,41,42)(36,43,46,45)(40,48,44,47);(1,5,6,8)(2,9,10,3)(4,15,13,16)(7,14,12,11)(17,22,23,26)(18,20,25,30)(19,27,28,32)(21,24,29,31)(33,34)(35,39)(36,40)(37,42)(38,41)(43,47)(44,46)(45,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=hi,d^2=hi,e^2=h,f^2=gh,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, 9067949);