#7658435

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

4309 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:

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

Polycyclic presentation:

2^9|c^2=h,d^2=g,e^2=gi,f^2=i,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, 7565631);