#8932196

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

4242 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=gh,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, 8839392);