#9863297

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

4105 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)(4,5)(6,15)(7,9)(10,12)(11,16)(13,14)(17,18,21,25)(19,24,27,22)(20,28,29,23)(26,30,32,31)(33,34)(35,44)(36,46)(37,41)(38,45)(39,40)(42,47)(43,48);(1,3,7,11)(2,8,9,16)(4,12,13,6)(5,10,14,15)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32)(34,40)(35,37)(42,46)(43,45);(2,9)(3,11)(4,13)(10,15)(17,20,21,29)(18,23,25,28)(19,26,27,32)(22,30,24,31)(33,35,39,37)(34,41,40,44)(36,43,47,45)(38,42,48,46);(1,4)(2,5)(3,12)(6,11)(7,13)(8,10)(9,14)(15,16)(17,20)(18,23)(19,26)(21,29)(22,30)(24,31)(25,28)(27,32)(33,36,39,47)(34,42,40,46)(35,45,37,43)(38,44,48,41);(1,5)(2,4)(3,10)(6,16)(7,14)(8,12)(9,13)(11,15)(17,19)(18,24)(20,26)(21,27)(22,25)(23,31)(28,30)(29,32)(33,37,39,35)(34,41,40,44)(36,43,47,45)(38,46,48,42);(1,6,7,12)(2,10,9,15)(3,4,11,13)(5,8,14,16)(17,18)(19,22)(20,23)(21,25)(24,27)(26,30)(28,29)(31,32)(33,38)(34,43)(35,42)(36,44)(37,46)(39,48)(40,45)(41,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=ghi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9770493);