#8603520

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

4510 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8510716);