#273407

order	512
nilpotency class	3
isoclinism	isoclinic class of #273307
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#61623	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1116 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fhi,c^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,g^c=gi,e^d=ei,f^d=fhi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180603);