#7659950

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

4511 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=h,e^2=gi,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, 7567146);