#7966196

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

4588 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)(9,10,15,17)(11,16,20,22)(12,13,21,18)(14,19,23,24)(28,36)(33,38)(35,37)(39,40);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,14)(13,19)(15,20)(17,22)(18,24)(21,23)(25,26,31,30)(27,34,29,32)(28,33,36,38)(35,40,37,39);(1,2,5,4)(3,6,8,7)(10,17)(11,20)(12,21)(19,24)(25,27,31,29)(26,32,30,34)(28,35,36,37)(33,39,38,40);(1,2,5,4)(3,6,8,7)(9,12)(10,18)(11,14)(13,17)(15,21)(16,24)(19,22)(20,23)(25,28,31,36)(26,33,30,38)(27,35,29,37)(32,39,34,40);(1,2,5,4)(3,6,8,7)(9,13,15,18)(10,12,17,21)(11,19,20,24)(14,22,23,16)(25,29,31,27)(26,32,30,34)(28,37,36,35)(33,39,38,40);(1,4,5,2)(3,7,8,6)(9,14)(10,19)(11,12)(13,16)(15,23)(17,24)(18,22)(20,21)(25,30,31,26)(27,34,29,32)(28,38,36,33)(35,40,37,39)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7873392);