#7710524

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	#2847	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4516 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)(13,21)(17,22)(18,23)(20,24)(26,31)(28,36)(29,39)(32,35);(1,2)(3,5)(4,6)(7,8)(9,10,14,15)(11,12,16,19)(13,17,21,22)(18,20,23,24)(26,31)(32,35)(33,38)(34,40);(1,3,4,7)(2,5,6,8)(10,15)(12,19)(17,22)(20,24)(25,26)(27,35)(28,38)(29,34)(30,31)(32,37)(33,36)(39,40);(9,11,14,16)(10,12,15,19)(13,18,21,23)(17,20,22,24)(25,27,30,37)(26,32,31,35)(28,29,36,39)(33,34,38,40);(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,22)(21,24)(25,28,30,36)(26,33,31,38)(27,29,37,39)(32,34,35,40);(9,13)(10,17)(11,18)(12,20)(14,21)(15,22)(16,23)(19,24)(25,29)(26,34)(27,36)(28,37)(30,39)(31,40)(32,38)(33,35)

Polycyclic presentation:

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

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, 7617720);