#372540

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
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

1196 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)(9,10,16,19,15,20,14,31)(11,21,23,34,24,33,12,17)(13,26,30,36,29,40,28,18)(22,35,27,39,38,32,25,37);(1,3,5,4)(2,6,7,8)(9,11)(10,17,20,34)(12,14)(13,22)(15,24)(16,23)(18,32,36,35)(19,21,31,33)(25,28)(26,37,40,39)(27,30)(29,38);(1,4,5,3)(2,6,7,8)(9,12,14,24,15,23,16,11)(10,17,19,21,20,34,31,33)(13,27,28,22,29,25,30,38)(18,35,26,39,36,32,40,37);(2,7)(6,8)(9,13,16,30,15,29,14,28)(10,18,31,40,20,36,19,26)(11,22,23,27,24,38,12,25)(17,32,33,39,34,35,21,37)

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,d^c=di,f^c=fi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 279736);