#8445738

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

4380 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=h,e^2=i,f^2=g,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, 8352934);