#8741586

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

4499 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)(9,10,14,11)(12,17,20,15)(13,16,22,18)(19,24,21,23)(25,26)(27,36)(28,33)(29,32)(30,34)(31,35)(37,38)(39,40);(2,7)(3,4)(10,11)(15,17)(16,18)(23,24)(25,27,31,29)(26,32,35,36)(28,37,34,39)(30,38,33,40);(1,3,6,4)(2,8,7,5)(10,11)(15,17)(16,18)(23,24)(25,28,31,34)(26,33,35,30)(27,37,29,39)(32,40,36,38);(1,4,6,3)(2,5,7,8)(9,11,14,10)(12,17,20,15)(13,16,22,18)(19,23,21,24)(25,28)(26,33)(27,37)(29,39)(30,35)(31,34)(32,40)(36,38);(1,5)(2,3)(4,7)(6,8)(9,12)(10,15)(11,17)(13,21)(14,20)(16,24)(18,23)(19,22)(25,29,31,27)(26,32,35,36)(28,39,34,37)(30,38,33,40);(9,13)(10,16)(11,18)(12,19)(14,22)(15,23)(17,24)(20,21)(25,30,31,33)(26,34,35,28)(27,38,29,40)(32,39,36,37)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8648782);