#483041

order	512
nilpotency class	4
isoclinism	isoclinic class of #11704
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#102
prime radical	C₂×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1124 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)(11,21,24,37)(12,18)(13,27)(14,20)(15,17)(16,23,31,40)(19,29)(22,35,25,38)(26,34)(28,32,39,33)(30,36);(2,5)(3,7)(9,11)(10,16)(12,22)(13,28)(14,25)(15,24)(17,31)(18,32)(19,35)(20,33)(21,36)(23,26)(27,37)(29,40)(30,39)(34,38);(10,17)(16,31)(18,20)(19,34)(21,37)(27,36)(32,33)(35,38);(9,12,15,14)(10,18,17,20)(11,22,24,25)(13,26,30,29)(16,32,31,33)(19,27,34,36)(21,35,37,38)(23,39,40,28)

Polycyclic presentation:

2^9|a^2=f,d^2=i,e^2=hi,f^2=gh,g^2=i,h^2=i,b^a=be,c^a=ci,e^a=eg,g^a=gi,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390237);