#9835870

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

4272 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)(17,18,23,22)(19,25,20,24)(21,26,29,30)(27,32,28,31)(34,38)(35,41)(40,43)(42,47);(1,2)(3,7)(4,8)(5,6)(9,15)(10,14)(11,13)(12,16)(17,19,23,20)(18,24,22,25)(21,28,29,27)(26,32,30,31)(33,34)(35,39)(36,40)(37,38)(41,45)(42,46)(43,44)(47,48);(2,6)(7,14)(8,11)(15,16)(33,35,37,41)(34,39,38,45)(36,42,44,47)(40,46,43,48);(1,2)(3,7)(4,8)(5,6)(9,15)(10,14)(11,13)(12,16)(17,20,23,19)(18,25,22,24)(21,28,29,27)(26,32,30,31)(33,35)(34,39)(36,42)(37,41)(38,45)(40,46)(43,48)(44,47);(1,3,5,10)(2,7,6,14)(4,12,13,9)(8,16,11,15)(17,21)(18,26)(19,27)(20,28)(22,30)(23,29)(24,31)(25,32)(33,36,37,44)(34,40,38,43)(35,42,41,47)(39,46,45,48);(1,4,5,13)(2,8,6,11)(3,9,10,12)(7,15,14,16)(17,22,23,18)(19,25,20,24)(21,26,29,30)(27,31,28,32)(33,34)(35,39)(36,43)(37,38)(40,44)(41,45)(42,48)(46,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=hi,d^2=ghi,e^2=h,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, 9743066);