#8371283

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

4521 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gh,d^2=ghi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8278479);