#7640222

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#40534	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4504 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)(3,6)(4,5)(7,8)(9,10,15,18)(11,16,12,20)(13,22,14,17)(19,24,21,23)(25,26,29,32)(27,30,34,38)(28,31,36,40)(33,37,35,39);(2,5)(6,8)(9,11,15,12)(10,16,18,20)(13,21,14,19)(17,24,22,23)(25,27)(26,30)(28,35)(29,34)(31,39)(32,38)(33,36)(37,40);(9,12,15,11)(10,16,18,20)(13,21,14,19)(17,23,22,24)(25,28)(26,31)(27,33)(29,36)(30,37)(32,40)(34,35)(38,39);(9,13,15,14)(10,17,18,22)(11,19,12,21)(16,23,20,24)(27,34)(30,38)(33,35)(37,39);(9,14,15,13)(10,17,18,22)(11,19,12,21)(16,24,20,23)(28,36)(31,40)(33,35)(37,39);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7547418);