#273334

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	#3489	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1077 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=i,d^2=hi,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, 180530);