#9717413

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

4120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=g,e^2=ghi,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, 9624609);