#7646360

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

4467 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)(9,10,15,12)(11,18,13,16)(14,20,22,17)(19,23,21,24)(26,31)(27,32)(29,37)(35,40);(1,3,5,8)(2,6,4,7)(9,11,15,13)(10,16,12,18)(14,19,22,21)(17,23,20,24);(9,10,15,12)(11,18,13,16)(14,17,22,20)(19,24,21,23)(25,26,30,31)(27,28,32,36)(29,33,37,38)(34,35,39,40);(1,2,5,4)(3,6,8,7)(9,12,15,10)(11,18,13,16)(14,20,22,17)(19,24,21,23)(25,27)(26,28)(29,34)(30,32)(31,36)(33,35)(37,39)(38,40);(9,13,15,11)(10,16,12,18)(14,21,22,19)(17,23,20,24)(25,28)(26,32)(27,31)(29,35)(30,36)(33,39)(34,38)(37,40);(1,4,5,2)(3,7,8,6)(9,14,15,22)(10,17,12,20)(11,19,13,21)(16,23,18,24)(25,29)(26,33)(27,34)(28,35)(30,37)(31,38)(32,39)(36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7553556);