#508030

order	512
nilpotency class	4
isoclinism	isoclinic class of #508016
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 ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1002 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 415226);