#7642813

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	#40396	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4453 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:

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

Polycyclic presentation:

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

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, 7550009);