#7726603

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

4426 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=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, 7633799);