#7766344

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
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4246 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=h,e^2=g,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, 7673540);