#7798048

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

4339 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)(9,10)(11,17)(12,16)(13,15)(14,19)(18,23)(20,22)(21,24)(25,26,31,33)(27,29,30,32)(28,34,36,40)(35,39,38,37);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,18)(13,19)(15,17)(16,21)(20,24)(22,23)(25,27,31,30)(26,32,33,29)(28,35,36,38)(34,37,40,39);(1,4,5,2)(3,7,8,6)(10,15)(14,17)(16,22)(21,23)(26,33)(27,30)(28,36)(37,39);(9,10)(11,14)(12,16)(13,15)(17,19)(18,21)(20,22)(23,24)(25,28,31,36)(26,34,33,40)(27,35,30,38)(29,39,32,37);(9,11)(10,14)(12,18)(13,19)(15,17)(16,21)(20,24)(22,23)(25,29,31,32)(26,27,33,30)(28,37,36,39)(34,38,40,35);(9,12,13,20)(10,16,15,22)(11,18,19,24)(14,21,17,23)(25,30,31,27)(26,32,33,29)(28,38,36,35)(34,37,40,39)

Polycyclic presentation:

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

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, 7705244);