#7726715

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

4349 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)(11,21)(12,17)(13,18)(14,19)(15,20)(16,22)(23,24)(25,26)(27,32)(28,29)(30,34)(31,37)(33,39)(35,36)(38,40);(1,3,5,8)(2,6,4,7)(9,11,15,22)(10,16,20,21)(12,13,19,24)(14,18,17,23)(25,27)(26,31)(28,35)(29,40)(30,37)(32,34)(33,36)(38,39);(11,22)(13,24)(16,21)(18,23)(25,27,30,37)(26,31,34,32)(28,38,39,35)(29,36,33,40);(1,2,5,4)(3,6,8,7)(9,12,15,19)(10,17,20,14)(11,13,22,24)(16,23,21,18)(25,27)(26,32)(28,35)(29,36)(30,37)(31,34)(33,40)(38,39);(1,2,5,4)(3,6,8,7)(9,13)(10,18)(11,19)(12,22)(14,21)(15,24)(16,17)(20,23)(25,28)(26,33)(27,35)(29,34)(30,39)(31,36)(32,40)(37,38);(1,4,5,2)(3,7,8,6)(9,14,15,17)(10,19,20,12)(11,23,22,18)(13,21,24,16)(25,29,30,33)(26,28,34,39)(27,36,37,40)(31,38,32,35)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=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, 7633911);