#7644198

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

4361 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:

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

Polycyclic presentation:

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

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, 7551394);