#9057909

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

4492 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8965105);