#7795768

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

4391 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=gh,e^2=gi,f^2=gh,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, 7702964);