#502872

order	512
nilpotency class	4
isoclinism	isoclinic class of #501058
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

893 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)(3,7)(4,5)(6,8)(9,10)(11,19)(12,15)(13,26)(14,27)(16,22)(17,21)(18,25)(20,31)(23,38)(24,39)(28,35)(29,34)(30,40)(32,37)(33,36);(2,5)(6,7)(10,15)(11,20)(13,24)(16,28)(18,27)(21,34)(23,36)(30,32);(9,11,21,37,14,23,24,35,12,20,34,40,25,36,13,22)(10,16,26,33,18,30,29,31,15,28,39,38,27,32,17,19);(1,3,4,8)(2,6,5,7)(10,15)(16,28)(17,29)(18,27)(19,31)(26,39)(30,32)(33,38)

Polycyclic presentation:

2^9|c^2=fh,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410068);