#7642571

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

4508 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)(10,15)(13,23)(16,24)(17,20)(26,27)(29,37)(31,33)(35,40);(1,2)(3,6)(4,5)(7,8)(9,10,14,15)(11,16,21,24)(12,17,22,20)(13,18,23,19)(25,26,30,27)(28,31,36,33)(29,32,37,34)(35,38,40,39);(10,15)(12,22)(16,24)(18,19)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(1,3,4,7)(2,6,5,8)(9,11,14,21)(10,16,15,24)(12,19,22,18)(13,20,23,17)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(9,12)(10,17)(11,19)(13,16)(14,22)(15,20)(18,21)(23,24)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(9,13)(10,18)(11,20)(12,16)(14,23)(15,19)(17,21)(22,24)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7549767);