#9923981

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

4504 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)(4,6)(7,8)(10,12)(14,15)(17,18,23,26)(19,27,30,20)(21,24,33,37)(22,35,36,25)(28,39,41,31)(29,32,43,40)(34,46,45,38)(42,44,48,47);(1,2,4,6)(3,5,8,7)(9,10,13,12)(11,15,16,14)(21,33)(24,37)(28,41)(31,39)(34,45)(38,46)(42,48)(44,47);(9,11,13,16)(10,14,12,15)(17,19,23,30)(18,20,26,27)(21,28,33,41)(22,29,36,43)(24,31,37,39)(25,32,35,40)(34,42,45,48)(38,44,46,47);(9,12,13,10)(11,15,16,14)(17,20)(18,19)(21,31)(22,32)(23,27)(24,28)(25,29)(26,30)(33,39)(34,44)(35,43)(36,40)(37,41)(38,42)(45,47)(46,48);(1,2)(3,7)(4,6)(5,8)(17,21,23,33)(18,24,26,37)(19,28,30,41)(20,31,27,39)(22,34,36,45)(25,38,35,46)(29,42,43,48)(32,44,40,47);(1,3)(2,5)(4,8)(6,7)(17,22,23,36)(18,25,26,35)(19,29,30,43)(20,32,27,40)(21,34,33,45)(24,38,37,46)(28,42,41,48)(31,44,39,47)

Polycyclic presentation:

2^9|a^2=h,b^2=g,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, 9831177);