#372561

order	512
nilpotency class	3
isoclinism	isoclinic class of #25440
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	D₄×C₂²⋊C₄
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1220 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)(3,8)(4,6)(5,7)(9,10)(11,18)(12,25,29,17)(13,30)(14,20)(15,21)(16,23)(19,31)(22,37,39,34)(24,33)(26,32,38,40)(27,35,28,36);(1,3,5,4)(2,6,7,8)(9,11,14,23,15,24,31,13)(10,16,19,18,21,30,20,33)(12,26,28,39,29,38,27,22)(17,34,36,32,25,37,35,40);(1,4,5,3)(2,6,7,8)(9,12,14,27,15,29,31,28)(10,17,19,35,21,25,20,36)(11,22,23,38,24,39,13,26)(16,32,18,34,30,40,33,37);(2,7)(6,8)(9,13,31,24,15,23,14,11)(10,18,20,16,21,33,19,30)(12,26,28,39,29,38,27,22)(17,34,36,32,25,37,35,40)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=e,d^2=fg,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,f^c=fi,g^c=gi,h^c=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, 279757);