#9717469

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

4108 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=di,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, 9624665);