#7659067

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

4558 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)(11,19)(12,17)(13,18)(14,15)(16,20)(21,23)(22,24);(1,2)(3,6)(4,5)(7,8)(25,26)(27,31)(28,32)(29,30)(33,39)(34,37)(35,40)(36,38);(10,15)(11,20)(17,24)(18,21)(26,30)(31,37)(32,38)(39,40);(9,11)(10,16)(12,21)(13,17)(14,20)(15,19)(18,22)(23,24)(25,26)(27,31)(28,32)(29,30)(33,39)(34,37)(35,40)(36,38);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,21)(13,16)(14,22)(15,24)(18,20)(19,23)(25,27,29,34)(26,31,30,37)(28,35,36,33)(32,40,38,39);(9,13)(10,18)(11,17)(12,19)(14,23)(15,21)(16,22)(20,24)(25,28,29,36)(26,32,30,38)(27,33,34,35)(31,39,37,40)

Polycyclic presentation:

2^9|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, 7566263);