#8445906

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

4520 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)(3,8,6,4)(9,10,16,19)(11,17,14,15)(12,22,13,18)(20,23,21,24)(25,26,29,30)(27,33,35,31)(28,32,37,36)(34,39,40,38);(1,3)(2,4)(5,6)(7,8)(9,11,16,14)(10,17,19,15)(12,21,13,20)(18,23,22,24)(26,30)(31,33)(32,36)(38,39);(9,12,16,13)(10,18,19,22)(11,20,14,21)(15,23,17,24)(25,27,29,35)(26,31,30,33)(28,34,37,40)(32,38,36,39);(1,4,5,8)(2,6,7,3)(9,13,16,12)(10,18,19,22)(11,21,14,20)(15,23,17,24)(25,26)(27,31)(28,36)(29,30)(32,37)(33,35)(34,39)(38,40);(1,3)(2,4)(5,6)(7,8)(9,14,16,11)(10,15,19,17)(12,21,13,20)(18,23,22,24);(1,4)(2,3)(5,8)(6,7)(9,15)(10,11)(12,23)(13,24)(14,19)(16,17)(18,20)(21,22)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=h,d^2=gh,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8353102);