#7652022

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

4393 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)(9,10,15,19)(11,20,21,16)(12,13,17,23)(14,18,24,22)(25,26,31,30)(27,32,29,33)(28,36,37,34)(35,40,38,39);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,14)(13,22)(15,21)(17,24)(18,23)(19,20)(25,27,31,29)(26,32,30,33)(28,35,37,38)(34,39,36,40);(11,21)(14,24)(16,20)(18,22)(25,27,31,29)(26,33,30,32)(28,35,37,38)(34,40,36,39);(9,12)(10,13)(11,14)(15,17)(16,22)(18,20)(19,23)(21,24)(25,28)(26,34)(27,35)(29,38)(30,36)(31,37)(32,39)(33,40);(1,4,5,2)(3,7,8,6)(9,13)(10,17)(11,22)(12,19)(14,20)(15,23)(16,24)(18,21)(25,29,31,27)(26,33,30,32)(28,35,37,38)(34,39,36,40);(9,14)(10,18)(11,12)(13,20)(15,24)(16,23)(17,21)(19,22)(25,30,31,26)(27,33,29,32)(28,36,37,34)(35,40,38,39)

Polycyclic presentation:

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

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, 7559218);