#7715660

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

4678 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei

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, 7622856);