#9040100

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

4470 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8947296);