#9726251

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

4266 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)(4,13)(5,10)(6,14)(7,9)(11,16)(12,15)(17,18)(19,20)(21,25)(22,24)(23,27)(26,28)(29,31)(30,32)(33,34)(35,44)(36,46)(37,41)(38,45)(39,40)(42,47)(43,48);(1,3,7,4)(2,8,9,13)(5,14,15,11)(6,12,16,10)(17,19)(18,23)(20,24)(21,29)(22,27)(25,28)(26,30)(31,32)(34,40)(35,37)(42,46)(43,45);(2,9)(3,4)(10,12)(11,14)(18,24)(20,23)(25,32)(28,31)(33,35,39,37)(34,41,40,44)(36,43,47,45)(38,42,48,46);(1,4,7,3)(2,8,9,13)(5,14,15,11)(6,10,16,12)(17,20,22,23)(18,19,24,27)(21,28,30,31)(25,26,32,29)(33,36,39,47)(34,42,40,46)(35,45,37,43)(38,44,48,41);(1,5,7,15)(2,10,9,12)(3,11,4,14)(6,8,16,13)(17,21)(18,25)(19,26)(20,28)(22,30)(23,31)(24,32)(27,29)(33,37,39,35)(34,41,40,44)(36,43,47,45)(38,46,48,42);(1,6,7,16)(2,11,9,14)(3,12,4,10)(5,13,15,8)(17,19)(18,23)(20,24)(21,29)(22,27)(25,28)(26,30)(31,32)(33,38)(34,43)(35,42)(36,44)(37,46)(39,48)(40,45)(41,47)

Polycyclic presentation:

2^9|b^2=gh,c^2=ghi,d^2=h,e^2=i,f^2=gh,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, 9633447);