#8492841

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
prime radical	#39128	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4488 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=gi,d^2=gh,e^2=i,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, 8400037);