#447857

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2563
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

635 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)(12,25)(13,27,30,36)(14,31)(15,21)(16,20)(17,23)(19,28,37,29)(22,38,40,32)(24,34)(26,35,39,33);(1,2)(3,7)(4,5)(6,8)(9,11,16,12,15,24,14,23)(10,17,31,34,21,25,20,18)(13,26,29,22,30,39,28,40)(19,35,27,32,37,33,36,38);(1,3,4,8)(2,6,5,7)(9,12,16,24,15,23,14,11)(10,18,31,17,21,34,20,25)(13,26,28,40,30,39,29,22)(19,33,36,32,37,35,27,38);(2,5)(6,7)(9,13,14,29,15,30,16,28)(10,19,20,27,21,37,31,36)(11,22,23,39,24,40,12,26)(17,32,18,33,25,38,34,35)

Polycyclic presentation:

2^9|a^2=efg,b^2=eg,c^2=fg,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 355053);