#7659904

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) = 256

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi,f^e=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7567100);