#8932159

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

4407 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)(4,8)(6,7)(10,12)(14,15)(18,23)(19,27)(21,34)(25,36)(29,31)(32,45)(38,42)(47,48);(2,7)(3,8)(9,10,13,12)(11,15,16,14)(18,23)(24,28)(25,36)(26,35)(37,39)(38,42)(40,46)(47,48);(1,3,6,8)(2,4,7,5)(9,11,13,16)(10,14,12,15)(17,18)(19,28)(20,25)(21,35)(22,23)(24,27)(26,34)(29,39)(30,38)(31,37)(32,46)(33,36)(40,45)(41,42)(43,47)(44,48);(1,4,6,5)(2,3,7,8)(9,12,13,10)(11,15,16,14)(17,19)(18,24)(20,31)(21,30)(22,27)(23,28)(25,39)(26,38)(29,33)(32,44)(34,41)(35,42)(36,37)(40,48)(43,45)(46,47);(1,2)(3,4)(5,8)(6,7)(17,20,22,33)(18,25,23,36)(19,29,27,31)(21,32,34,45)(24,37,28,39)(26,40,35,46)(30,43,41,44)(38,47,42,48);(1,5,6,4)(2,8,7,3)(17,21,22,34)(18,26,23,35)(19,30,27,41)(20,32,33,45)(24,38,28,42)(25,40,36,46)(29,43,31,44)(37,47,39,48)

Polycyclic presentation:

2^9|b^2=hi,c^2=g,d^2=g,e^2=gi,f^2=h,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, 8839355);