#273315

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	#3488	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1101 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)(3,7,8,5)(4,6)(9,10)(11,16)(12,24)(13,15)(14,19)(17,25)(18,28)(20,31)(21,36)(22,29)(23,32)(26,33)(27,34)(30,35)(37,39)(38,40);(1,3)(2,5)(4,8)(6,7)(10,15)(11,20)(13,28)(17,24)(18,19)(21,35)(22,23)(25,26)(29,31)(33,34)(36,39)(37,38);(1,2)(3,5)(4,6)(7,8)(9,11,14,23)(10,16,19,32)(12,21,27,37)(13,22,28,20)(15,29,18,31)(17,30,33,40)(24,36,34,39)(25,35,26,38);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17)(11,21)(13,26)(14,27)(15,24)(16,30)(18,34)(19,33)(20,35)(22,38)(23,37)(25,28)(29,36)(31,39)(32,40)

Polycyclic presentation:

2^9|a^2=f,c^2=i,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, 180511);