#9923980

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

4502 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^2=i,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, 9831176);