#8260850

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

4653 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=ghi,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8168046);