#481587

order	512
nilpotency class	4
isoclinism	isoclinic class of #11704
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#102
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

819 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gh,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,d^a=di,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, 388783);