#7658226

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

4580 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)(9,10)(11,18)(12,17)(13,19)(14,15)(16,21)(20,24)(22,23)(28,36)(31,37)(34,40)(38,39);(1,2)(3,6)(4,5)(7,8)(10,15)(12,22)(16,18)(19,20)(25,26)(27,32)(28,37)(29,33)(30,35)(31,36)(34,38)(39,40);(9,11)(10,16)(12,20)(13,17)(14,21)(15,18)(19,22)(23,24)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,40)(37,39);(9,12)(10,17)(11,20)(13,16)(14,22)(15,23)(18,24)(19,21)(25,28)(26,31)(27,34)(29,36)(30,38)(32,39)(33,37)(35,40);(1,3,4,7)(2,6,5,8)(9,11)(10,18)(12,20)(13,23)(14,21)(15,16)(17,24)(19,22)(25,27)(26,32)(28,34)(29,35)(30,33)(31,39)(36,40)(37,38);(9,13,14,24)(10,19,15,20)(11,17,21,23)(12,16,22,18)(25,26)(27,30)(28,31)(29,33)(32,35)(34,38)(36,37)(39,40)

Polycyclic presentation:

2^9|e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7565422);