#9923926

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
prime radical	#45838	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3941 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:

(1,2,7,9)(3,10,4,5)(6,8,15,16)(11,14,12,13)(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);(1,3,7,4)(2,5,9,10)(6,11,15,12)(8,13,16,14)(17,19)(18,24)(20,30)(21,28)(22,31)(23,29)(25,32)(26,27)(35,41)(38,45)(42,47)(46,48);(18,25)(19,29)(20,28)(27,31)(33,34,37,40)(35,38,41,45)(36,39,44,43)(42,46,47,48);(1,4,7,3)(2,5,9,10)(6,12,15,11)(8,13,16,14)(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)(2,3)(4,9)(6,14)(7,10)(8,12)(11,16)(13,15)(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)(2,8)(3,11)(4,12)(5,13)(7,15)(9,16)(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=hi,d^2=gi,e^2=h,f^2=h,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, 9831122);