#8458085

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

4618 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gi,d^2=h,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8365281);