#8979718

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

4281 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,29)(30,32)(34,39)(35,42)(41,47)(43,45);(2,6)(7,9)(8,12)(14,15)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,30)(29,32)(33,34,38,39)(35,40,42,37)(36,41,44,47)(43,48,45,46);(2,6)(3,10)(4,13)(14,15)(18,22)(19,21)(20,28)(31,32)(35,42)(36,44)(37,40)(41,47);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,20,23,28)(18,25,22,29)(19,27,21,30)(24,31,26,32)(33,35)(34,40)(36,45)(37,39)(38,42)(41,46)(43,44)(47,48);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,21,23,19)(18,26,22,24)(20,27,28,30)(25,31,29,32)(33,36,38,44)(34,41,39,47)(35,43,42,45)(37,46,40,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,22,23,18)(19,26,21,24)(20,29,28,25)(27,32,30,31)(33,37,38,40)(34,35,39,42)(36,46,44,48)(41,43,47,45)

Polycyclic presentation:

2^9|b^2=ghi,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8886914);