#8621572

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

4694 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=g,d^2=gh,e^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8528768);