#8602057

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

4675 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=hi,d^2=h,e^2=gh,f^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, 8509253);