#9726242

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

4120 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)(4,9)(5,13)(6,7)(10,15)(11,14)(12,16)(17,18)(19,24)(20,29)(21,26)(22,30)(23,25)(27,32)(28,31)(34,40)(35,38)(36,42)(45,47);(2,7)(4,12)(8,14)(10,13)(17,19,23,20)(18,24,25,29)(21,30,31,27)(22,28,32,26)(33,34)(35,43)(36,37)(38,41)(39,40)(42,48)(44,45)(46,47);(2,7)(3,11)(9,16)(10,13)(18,25)(19,20)(26,28)(27,30)(33,35,39,38)(34,41,40,43)(36,46,42,44)(37,45,48,47);(1,3,6,11)(2,8,7,14)(4,10,12,13)(5,9,15,16)(17,20,23,19)(18,24,25,29)(21,30,31,27)(22,26,32,28)(33,36,39,42)(34,37,40,48)(35,44,38,46)(41,47,43,45);(1,4)(2,9)(3,10)(5,14)(6,12)(7,16)(8,15)(11,13)(17,21,23,31)(18,26,25,28)(19,27,20,30)(22,24,32,29)(33,37)(34,42)(35,45)(36,40)(38,47)(39,48)(41,44)(43,46);(1,5,6,15)(2,10,7,13)(3,9,11,16)(4,8,12,14)(17,22,23,32)(18,27,25,30)(19,28,20,26)(21,29,31,24)(33,38,39,35)(34,41,40,43)(36,46,42,44)(37,47,48,45)

Polycyclic presentation:

2^9|b^2=gh,c^2=ghi,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9633438);