#478504

order	512
nilpotency class	3
isoclinism	isoclinic class of #476297
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

800 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=ef,c^2=i,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,g^b=gi,d^c=de,g^c=gi,h^c=hi

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, 385700);