#368973

order	512
nilpotency class	3
isoclinism	isoclinic class of #24805
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1188 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)(11,15)(12,16,23,33)(13,18)(14,17)(19,27,35,39)(20,28)(21,29)(22,30,37,40)(24,32,25,34)(26,31)(36,38);(2,5)(6,7)(9,11)(10,15)(12,22,23,37)(13,21)(14,20)(16,30,33,40)(17,28)(18,29)(19,24,35,25)(26,36)(27,32,39,34)(31,38);(9,12,13,25,14,23,26,24)(10,16,31,32,17,33,18,34)(11,19,21,37,20,35,36,22)(15,27,38,30,28,39,29,40);(1,3,4,8)(2,6,5,7)(11,20)(15,28)(19,35)(21,36)(22,37)(27,39)(29,38)(30,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276169);