#474267

order	512
nilpotency class	3
isoclinism	isoclinic class of #458126
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2572
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

569 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)(3,6)(4,5)(7,8)(9,10,15,20,13,19,14,18)(11,21,30,33,23,16,24,32)(12,25,29,35,27,17,28,36)(22,31,39,40,38,37,26,34);(2,5)(3,7)(9,11)(10,16)(12,26,27,39)(13,23)(14,24)(15,30)(17,34,25,40)(18,33)(19,21)(20,32)(22,28,38,29)(31,35,37,36);(1,3,4,7)(2,6,5,8)(9,12,13,27)(10,17,19,25)(11,22,23,38)(14,29,15,28)(16,31,21,37)(18,36,20,35)(24,39,30,26)(32,34,33,40);(3,7)(6,8)(10,18,19,20)(11,23)(12,27)(16,32,21,33)(17,35,25,36)(24,30)(28,29)(31,40,37,34)

Polycyclic presentation:

2^9|a^2=eg,b^2=efgh,c^2=e,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=di,d^c=de,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381463);