#8504821

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

4485 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)(3,8)(4,7)(5,6)(9,10,15,14)(11,16,22,21)(12,18,13,17)(19,24,20,23)(25,26,32,35)(27,28,36,38)(29,33,37,40)(30,31,34,39);(2,6)(3,7)(9,11,15,22)(10,16,14,21)(12,19,13,20)(17,23,18,24)(25,27,32,36)(26,28,35,38)(29,39,37,31)(30,40,34,33);(9,12,15,13)(10,17,14,18)(11,19,22,20)(16,23,21,24)(25,28)(26,27)(29,34)(30,37)(31,33)(32,38)(35,36)(39,40);(2,6)(4,8)(9,11)(10,16)(12,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,29,32,37)(26,33,35,40)(27,31,36,39)(28,34,38,30);(1,3,5,7)(2,4,6,8)(9,13,15,12)(10,18,14,17)(11,20,22,19)(16,24,21,23)(25,30,32,34)(26,31,35,39)(27,33,36,40)(28,37,38,29);(1,4)(2,7)(3,6)(5,8)(9,14,15,10)(11,21,22,16)(12,18,13,17)(19,24,20,23)(25,31)(26,34)(27,29)(28,33)(30,35)(32,39)(36,37)(38,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=hi,d^2=i,e^2=gh,f^2=hi,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, 8412017);