#8471117

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

4571 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)(4,6)(5,7)(9,10,15,12)(11,16,22,19)(13,23,21,17)(14,18,20,24)(25,26,31,35)(27,36,37,32)(28,33,38,40)(29,30,34,39);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,19)(13,20)(14,21)(15,22)(17,18)(23,24)(27,37)(28,38)(32,36)(33,40);(9,10,15,12)(11,19,22,16)(13,17,21,23)(14,18,20,24)(25,27)(26,32)(28,34)(29,38)(30,33)(31,37)(35,36)(39,40);(2,7)(6,8)(9,12,15,10)(11,19,22,16)(13,23,21,17)(14,18,20,24)(25,28,31,38)(26,33,35,40)(27,34,37,29)(30,36,39,32);(1,4,5,3)(2,8,7,6)(9,13,15,21)(10,17,12,23)(11,20,22,14)(16,18,19,24)(25,29,31,34)(26,30,35,39)(27,38,37,28)(32,33,36,40);(1,2)(3,6)(4,8)(5,7)(9,14,15,20)(10,18,12,24)(11,21,22,13)(16,17,19,23)(25,30)(26,34)(27,33)(28,32)(29,35)(31,39)(36,38)(37,40)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=hi,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, 8378313);