#7970284

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

4545 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)(9,10,14,15)(11,17,19,12)(13,16,21,22)(18,23,24,20)(25,26,30,32)(27,35,36,31)(28,39,38,33)(29,34,37,40);(1,3,5,8)(2,6,4,7)(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,24)(22,23)(25,27,30,36)(26,31,32,35)(28,37,38,29)(33,40,39,34);(1,2,5,4)(3,6,8,7)(10,15)(12,17)(16,22)(20,23)(26,32)(27,36)(28,38)(34,40);(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,23)(22,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,36)(37,40)(38,39);(1,2,5,4)(3,6,8,7)(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,24)(22,23)(25,28)(26,33)(27,37)(29,36)(30,38)(31,40)(32,39)(34,35);(1,4,5,2)(3,7,8,6)(9,13,14,21)(10,16,15,22)(11,18,19,24)(12,20,17,23)(25,29,30,37)(26,34,32,40)(27,38,36,28)(31,39,35,33)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,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, 7877480);