#8603525

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

4849 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=i,d^2=gh,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, 8510721);