#7857716

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

4341 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)(9,10,14,16)(11,15,19,22)(12,20,21,17)(13,18,23,24)(25,26)(27,37)(28,29)(30,34)(31,35)(32,36)(33,40)(38,39);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,27)(26,33)(28,35)(29,39)(30,38)(31,34)(32,40)(36,37);(1,2,5,4)(3,6,8,7)(10,16)(11,19)(17,20)(18,24)(25,28,32,30)(26,29,36,34)(27,38,40,35)(31,37,39,33);(1,2,5,4)(3,6,8,7)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,29,32,34)(26,28,36,30)(27,39,40,31)(33,35,37,38);(1,2,5,4)(3,6,8,7)(9,12,14,21)(10,17,16,20)(11,18,19,24)(13,22,23,15)(25,30,32,28)(26,34,36,29)(27,38,40,35)(31,37,39,33);(1,4,5,2)(3,7,8,6)(9,13)(10,18)(11,17)(12,15)(14,23)(16,24)(19,20)(21,22)(25,31,32,39)(26,35,36,38)(27,34,40,29)(28,37,30,33)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7764912);