#244083

order	512
nilpotency class	3
isoclinism	isoclinic class of #238001
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#400
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

566 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=efi,c^2=e,e^2=h,g^2=i,b^a=be,c^a=cf,g^a=gh,c^b=cg,e^b=eh,f^b=fh,g^b=gi,d^c=dh,g^c=ghi

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, 151279);