#273196

order	512
nilpotency class	3
isoclinism	isoclinic class of #273179
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

932 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180392);