#9873181

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

4132 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)(3,7)(4,11)(5,6)(8,13)(9,16)(10,14)(12,15)(17,18)(19,28)(20,22)(21,24)(23,25)(26,27)(29,32)(30,31)(34,38)(36,44)(39,41)(42,48);(1,2)(3,7)(4,8)(5,6)(9,15)(10,14)(11,13)(12,16)(17,19,23,21)(18,24,25,28)(20,29,27,31)(22,30,26,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(2,6)(7,14)(8,11)(15,16)(18,25)(20,27)(24,28)(29,31)(33,34,37,38)(35,39,43,41)(36,40,44,45)(42,46,48,47);(1,2)(3,7)(4,8)(5,6)(9,15)(10,14)(11,13)(12,16)(17,20,23,27)(18,26,25,22)(19,29,21,31)(24,32,28,30)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,3,5,10)(2,7,6,14)(4,12,13,9)(8,16,11,15)(17,21,23,19)(18,24,25,28)(20,31,27,29)(22,30,26,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,4)(2,8)(3,9)(5,13)(6,11)(7,15)(10,12)(14,16)(17,22,23,26)(18,27,25,20)(19,30,21,32)(24,31,28,29)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|b^2=hi,c^2=ghi,d^2=hi,e^2=gi,f^2=g,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, 9780377);