#7720336

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

4414 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=g,e^2=g,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=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, 7627532);