#7659530

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	#52280	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4326 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7566726);