#8505770

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

4447 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8412966);