#9712305

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

4106 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)(3,11)(4,9)(5,13)(6,7)(8,12)(10,15)(14,16)(17,18)(19,24)(20,22)(21,26)(23,25)(27,29)(28,31)(30,32)(33,34,37,40)(35,41,43,38)(36,39,45,48)(42,47,44,46);(2,7)(4,14)(8,11)(10,13)(17,19,23,20)(18,24,25,22)(21,28,30,27)(26,31,32,29)(33,35)(34,38)(36,44)(37,43)(39,47)(40,41)(42,45)(46,48);(2,7)(3,12)(9,16)(10,13)(18,25)(19,20)(26,32)(27,28)(33,35,37,43)(34,38,40,41)(36,42,45,44)(39,46,48,47);(1,3,6,12)(2,8,7,11)(4,13,14,10)(5,9,15,16)(17,20,23,19)(18,22,25,24)(21,28,30,27)(26,31,32,29)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,45)(47,48);(1,4,6,14)(2,9,7,16)(3,13,12,10)(5,8,15,11)(17,21)(18,26)(19,27)(20,28)(22,31)(23,30)(24,29)(25,32)(33,36,37,45)(34,39,40,48)(35,42,43,44)(38,46,41,47);(1,5,6,15)(2,10,7,13)(3,9,12,16)(4,11,14,8)(17,22)(18,19)(20,25)(21,29)(23,24)(26,28)(27,32)(30,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=ghi,d^2=gi,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,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, 9619501);