#7715663

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4044 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,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, 7622859);