#9923954

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

4189 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=i,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9831150);