#9923894

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

3871 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)(3,6)(4,7)(5,13)(8,10)(9,16)(11,12)(14,15)(17,18)(19,23)(20,29)(21,25)(22,26)(24,28)(27,32)(30,31)(33,34,38,36)(35,39,43,41)(37,40,45,44)(42,46,48,47);(1,3,8,7)(2,6,10,4)(5,11,16,15)(9,14,13,12)(19,27)(21,30)(23,32)(25,31)(34,36)(39,41)(40,44)(46,47);(1,4,8,6)(2,3,10,7)(5,14,16,12)(9,15,13,11)(17,19,22,27)(18,23,26,32)(20,21,28,30)(24,31,29,25)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(1,5,8,16)(2,9,10,13)(3,11,7,15)(4,12,6,14)(17,20,22,28)(18,24,26,29)(19,21,27,30)(23,31,32,25);(1,6,8,4)(2,3,10,7)(5,14,16,12)(9,11,13,15)(17,21)(18,25)(19,28)(20,27)(22,30)(23,24)(26,31)(29,32)(33,36,38,34)(35,41,43,39)(37,40,45,44)(42,46,48,47);(1,7,8,3)(2,6,10,4)(5,15,16,11)(9,14,13,12)(17,18)(19,23)(20,24)(21,31)(22,26)(25,30)(27,32)(28,29)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,d^2=i,e^2=gh,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, 9831090);