#9923973

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

4361 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)(3,5)(4,8)(6,9)(7,10)(11,13)(12,14)(15,16)(17,18)(19,27)(20,29)(21,26)(22,30)(23,25)(24,28)(31,32)(33,34)(35,38)(36,43)(37,40)(39,44)(41,45)(42,48)(46,47);(1,3,7,4)(2,8,10,5)(6,11,15,12)(9,14,16,13)(17,19,23,28)(18,24,25,27)(20,21,30,31)(22,26,29,32)(35,41)(38,45)(42,47)(46,48);(18,25)(19,28)(20,30)(26,32)(33,34,37,40)(35,38,41,45)(36,39,44,43)(42,46,47,48);(1,4,7,3)(2,8,10,5)(6,12,15,11)(9,14,16,13)(17,20)(18,22)(19,21)(23,30)(24,26)(25,29)(27,32)(28,31)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,5,7,8)(2,3,10,4)(6,14,15,13)(9,12,16,11)(17,21)(18,26)(19,20)(22,24)(23,31)(25,32)(27,29)(28,30)(33,35,37,41)(34,38,40,45)(36,42,44,47)(39,46,43,48);(1,6,7,15)(2,9,10,16)(3,11,4,12)(5,13,8,14)(17,22)(18,20)(19,26)(21,24)(23,29)(25,30)(27,31)(28,32)(33,36,37,44)(34,39,40,43)(35,42,41,47)(38,46,45,48)

Polycyclic presentation:

2^9|b^2=g,c^2=hi,d^2=gi,e^2=gh,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, 9831169);