#7812104

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

4382 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)(9,10,15,13)(11,19,20,16)(12,17,22,23)(14,18,21,24)(25,26)(27,37)(28,34)(29,40)(30,36)(31,33)(32,39)(35,38);(1,3,5,8)(2,6,4,7)(9,11,15,20)(10,16,13,19)(12,21,22,14)(17,18,23,24)(25,27)(26,32)(28,35)(29,34)(30,38)(31,39)(33,37)(36,40);(11,20)(14,21)(16,19)(18,24)(26,33)(28,36)(32,37)(35,40);(1,2,5,4)(3,6,8,7)(9,12,15,22)(10,17,13,23)(11,21,20,14)(16,18,19,24)(25,28,31,36)(26,34,33,30)(27,35,39,40)(29,37,38,32);(1,2,5,4)(3,6,8,7)(9,13,15,10)(11,19,20,16)(12,17,22,23)(14,24,21,18)(25,29,31,38)(26,35,33,40)(27,34,39,30)(28,37,36,32);(1,4,5,2)(3,7,8,6)(9,14,15,21)(10,18,13,24)(11,22,20,12)(16,17,19,23)(25,30,31,34)(26,36,33,28)(27,38,39,29)(32,40,37,35)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7719300);