#7798082

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

4531 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=h,d^2=gi,e^2=i,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,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, 7705278);