#372539

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
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

1206 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)(9,10,16,19,15,20,14,30)(11,21,23,33,24,32,12,17)(13,26,29,34,28,39,27,18)(22,36,37,40,38,31,25,35);(1,3,5,4)(2,6,7,8)(9,11)(10,17,20,33)(12,14)(13,22)(15,24)(16,23)(18,31,34,36)(19,21,30,32)(25,27)(26,35,39,40)(28,38)(29,37);(1,4,5,3)(2,6,7,8)(9,12,14,24,15,23,16,11)(10,17,19,21,20,33,30,32)(13,25,27,38,28,37,29,22)(18,31,26,35,34,36,39,40);(2,7)(6,8)(9,13,16,29,15,28,14,27)(10,18,30,39,20,34,19,26)(11,22,23,37,24,38,12,25)(17,31,32,40,33,36,21,35)

Polycyclic presentation:

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

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, 279735);