#508172

order	512
nilpotency class	4
isoclinism	isoclinic class of #508022
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

880 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)(11,19)(12,15)(13,25)(14,28)(16,21)(17,27)(18,24)(20,35)(22,37)(23,38)(26,34)(29,33)(30,36)(31,40)(32,39);(2,5)(6,8)(10,15)(13,23)(18,28)(19,35)(21,36)(26,39)(27,40)(33,37);(3,7)(6,8)(9,11)(10,16)(12,20)(13,26)(14,29)(15,30)(17,33)(18,34)(19,25)(21,27)(22,24)(23,39)(28,32)(31,37)(35,38)(36,40);(1,3,4,7)(2,6,5,8)(11,20)(16,30)(19,35)(21,36)(22,29)(26,39)(32,34)(33,37)

Polycyclic presentation:

2^9|d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=de,f^c=fh,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, 415368);