#8926297

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

4404 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)(3,5)(4,8)(6,7)(11,17)(16,23)(18,21)(19,24)(26,30)(31,36)(32,37)(38,40);(2,7)(3,8)(9,10,14,15)(11,16,17,23)(12,20,22,13)(18,19,21,24)(28,35)(32,37)(33,39)(38,40);(1,3,6,8)(2,4,7,5)(10,15)(13,20)(16,23)(19,24)(25,26)(27,31)(28,32)(29,30)(33,38)(34,36)(35,37)(39,40);(1,4,6,5)(2,3,7,8)(9,11)(10,16)(12,21)(13,19)(14,17)(15,23)(18,22)(20,24)(25,27,29,34)(26,31,30,36)(28,33,35,39)(32,38,37,40);(1,2)(3,4)(5,8)(6,7)(9,12)(10,13)(11,18)(14,22)(15,20)(16,24)(17,21)(19,23)(25,28)(26,32)(27,33)(29,35)(30,37)(31,38)(34,39)(36,40);(1,5,6,4)(2,8,7,3)(9,13)(10,12)(11,19)(14,20)(15,22)(16,21)(17,24)(18,23)(25,26)(27,31)(28,32)(29,30)(33,38)(34,36)(35,37)(39,40)

Polycyclic presentation:

2^9|b^2=gi,c^2=g,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8833493);