#8581804

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

4325 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)(3,7)(9,10,14,16)(11,15,20,18)(12,17,22,24)(13,19,23,21)(25,26,31,28)(27,29,32,34)(30,33,38,36)(35,37,39,40);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,21)(13,17)(14,20)(16,18)(19,22)(23,24)(25,27)(26,29)(28,34)(30,35)(31,32)(33,37)(36,40)(38,39);(1,2)(3,6)(4,5)(7,8)(10,16)(11,20)(17,24)(19,21)(25,26,31,28)(27,34,32,29)(30,33,38,36)(35,40,39,37);(1,2)(3,6)(4,5)(7,8)(9,12,14,22)(10,17,16,24)(11,19,20,21)(13,15,23,18)(25,28,31,26)(27,34,32,29)(30,33,38,36)(35,37,39,40);(1,3,4,7)(2,6,5,8)(9,11)(10,18)(12,19)(13,17)(14,20)(15,16)(21,22)(23,24)(25,29,31,34)(26,32,28,27)(30,37,38,40)(33,39,36,35);(9,13)(10,19)(11,17)(12,18)(14,23)(15,22)(16,21)(20,24)(25,30,31,38)(26,33,28,36)(27,35,32,39)(29,37,34,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=hi,d^2=gi,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=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, 8489000);