#9040093

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	C₂×#535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4478 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=gi,e^2=gi,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, 8947289);