#8471145

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

4556 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)(3,5)(4,7)(6,8)(9,10,15,12)(11,16,22,19)(13,23,21,17)(14,18,20,24)(25,26,31,34)(27,32,29,28)(30,33,39,38)(35,40,37,36);(2,3)(6,7)(9,11)(10,16)(12,19)(13,20)(14,21)(15,22)(17,18)(23,24)(25,27,31,29)(26,28,34,32)(30,35,39,37)(33,36,38,40);(1,2,5,3)(4,6,8,7)(9,10,15,12)(11,19,22,16)(13,17,21,23)(14,18,20,24)(25,28,31,32)(26,27,34,29)(30,36,39,40)(33,35,38,37);(1,3,5,2)(4,7,8,6)(9,12,15,10)(11,19,22,16)(13,23,21,17)(14,18,20,24)(25,26)(27,28)(29,32)(30,38)(31,34)(33,39)(35,40)(36,37);(1,4)(2,6)(3,7)(5,8)(9,13,15,21)(10,17,12,23)(11,20,22,14)(16,18,19,24)(25,29,31,27)(26,32,34,28)(30,37,39,35)(33,40,38,36);(9,14,15,20)(10,18,12,24)(11,21,22,13)(16,17,19,23)(25,30)(26,33)(27,35)(28,36)(29,37)(31,39)(32,40)(34,38)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=ghi,d^2=i,e^2=gi,f^2=hi,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, 8378341);