#7783224

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

4549 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)(11,19)(12,16)(13,17)(14,18)(15,20)(21,24)(22,23)(25,26,31,28)(27,32,35,36)(29,39,38,33)(30,34,37,40);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,21)(13,22)(14,20)(16,17)(18,19)(23,24)(25,27)(26,32)(28,36)(29,37)(30,38)(31,35)(33,34)(39,40);(1,2,5,4)(3,6,8,7)(11,20)(13,21)(15,19)(17,24)(26,28)(27,35)(30,37)(33,39);(1,4,5,2)(3,7,8,6)(9,12,14,22)(10,16,18,23)(11,21,20,13)(15,17,19,24)(25,28,31,26)(27,36,35,32)(29,33,38,39)(30,40,37,34);(9,10)(11,15)(12,23)(13,17)(14,18)(16,22)(19,20)(21,24)(25,29,31,38)(26,33,28,39)(27,37,35,30)(32,34,36,40);(9,13,14,21)(10,17,18,24)(11,22,20,12)(15,16,19,23)(25,30,31,37)(26,34,28,40)(27,38,35,29)(32,33,36,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=ghi,d^2=gh,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,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, 7690420);