#7642182

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
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4380 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)(9,10,14,15)(11,18,17,13)(12,16,20,22)(19,24,23,21)(28,36)(32,37)(34,40)(38,39);(1,2)(3,6)(4,5)(7,8)(12,20)(16,22)(19,23)(21,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,3,4,7)(2,6,5,8)(10,15)(11,17)(16,22)(19,23)(26,30)(31,33)(32,37)(38,39);(9,11,14,17)(10,13,15,18)(12,19,20,23)(16,21,22,24)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(9,12,14,20)(10,16,15,22)(11,19,17,23)(13,21,18,24)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(9,13)(10,17)(11,15)(12,21)(14,18)(16,23)(19,22)(20,24)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40)

Polycyclic presentation:

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

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, 7549378);