#7717340

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

4409 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=hi,f^2=g,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, 7624536);