#8932547

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,d^2=gi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8839743);