#362973

order	512
nilpotency class	3
isoclinism	isoclinic class of #24278
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×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1040 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:

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

Polycyclic presentation:

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

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, 270169);