#7656920

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

4561 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=ghi,d^2=g,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

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, 7564116);