#8463683

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

4465 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=g,e^2=ghi,f^2=ghi,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, 8370879);