#7655330

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

4442 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)(9,10,15,18)(11,19,14,16)(12,17,23,13)(20,24,22,21)(26,31)(27,33)(34,39)(35,40);(1,2)(3,7)(4,8)(5,6)(9,11,15,14)(10,16,18,19)(12,22,23,20)(13,21,17,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,37)(36,40)(38,39);(1,3,5,4)(2,7,6,8)(9,12)(10,17)(11,20)(13,18)(14,22)(15,23)(16,21)(19,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,37)(36,40)(38,39);(1,3,5,4)(2,7,6,8)(9,13,15,17)(10,12,18,23)(11,21,14,24)(16,22,19,20)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,37,39,36);(1,4,5,3)(2,8,6,7)(9,12)(10,17)(11,22)(13,18)(14,20)(15,23)(16,24)(19,21)(25,28,30,32)(26,33,31,27)(29,37,38,36)(34,35,39,40);(9,14,15,11)(10,16,18,19)(12,22,23,20)(13,24,17,21)(25,29)(26,34)(27,35)(28,36)(30,38)(31,39)(32,37)(33,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=ghi,d^2=g,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi

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, 7562526);