#7783246

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

4567 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)(9,10)(11,17)(12,22)(13,18)(14,16)(15,19)(20,21)(23,24)(25,26,31,35)(27,34,29,32)(28,33,39,30)(36,37,38,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,20)(13,21)(14,19)(16,17)(18,23)(22,24)(25,27,31,29)(26,32,35,34)(28,36,39,38)(30,37,33,40);(10,16)(11,19)(13,22)(20,23)(25,28,31,39)(26,30,35,33)(27,36,29,38)(32,37,34,40);(1,3,5,4)(2,6,8,7)(9,11)(10,17)(12,23)(13,21)(14,19)(15,16)(18,20)(22,24)(25,28)(26,33)(27,36)(29,38)(30,35)(31,39)(32,40)(34,37);(9,12,14,18)(10,13,16,22)(11,20,19,23)(15,21,17,24)(25,29,31,27)(26,34,35,32)(28,38,39,36)(30,40,33,37);(1,4,5,3)(2,7,8,6)(9,13)(10,18)(11,21)(12,16)(14,22)(15,23)(17,20)(19,24)(25,30,31,33)(26,28,35,39)(27,37,29,40)(32,36,34,38)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=hi,d^2=gh,e^2=h,f^2=gi,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, 7690442);