#7726604

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

4410 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)(11,16)(12,14)(13,22)(15,17)(18,23)(19,21)(20,24)(25,26,31,30)(27,35,36,32)(28,33,37,38)(29,34,39,40);(1,2,5,6)(3,7,4,8)(9,11,15,13)(10,16,17,22)(12,18,20,21)(14,23,24,19)(25,27,31,36)(26,32,30,35)(28,29,37,39)(33,40,38,34);(10,17)(11,13)(12,20)(19,23)(27,36)(29,39)(32,35)(34,40);(1,3,5,4)(2,7,6,8)(9,12)(10,14)(11,18)(13,21)(15,20)(16,23)(17,24)(19,22)(25,28,31,37)(26,33,30,38)(27,29,36,39)(32,40,35,34);(1,3,5,4)(2,7,6,8)(9,13,15,11)(10,16,17,22)(12,21,20,18)(14,23,24,19)(25,29)(26,34)(27,37)(28,36)(30,40)(31,39)(32,33)(35,38);(1,4,5,3)(2,8,6,7)(9,14)(10,12)(11,19)(13,23)(15,24)(16,21)(17,20)(18,22)(25,30,31,26)(27,32,36,35)(28,38,37,33)(29,40,39,34)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7633800);