#376811

order	512
nilpotency class	3
isoclinism	isoclinic class of #24412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

977 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)(4,6)(5,8)(9,10,16,21,14,20,15,34)(11,22,26,36,25,17,35,37)(12,24,29,38,27,40,28,18)(13,30,33,39,31,19,32,23);(1,3,4,8)(2,5,6,7)(9,11,14,25)(10,17,20,22)(12,23,27,39)(13,24,31,40)(15,35,16,26)(18,33,38,32)(19,29,30,28)(21,37,34,36);(9,12,15,28,14,27,16,29)(10,18,21,24,20,38,34,40)(11,23,26,30,25,39,35,19)(13,17,32,36,31,22,33,37);(2,6)(5,7)(9,13,16,33,14,31,15,32)(10,19,21,23,20,30,34,39)(11,24,26,38,25,40,35,18)(12,17,29,37,27,22,28,36)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 284007);