#9980369

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

4040 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=i,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dghi,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9887565);