#365868

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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	#2361
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

786 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)(3,7)(4,5)(6,8)(9,10,15,20,16,22,14,21)(11,23,25,34,26,18,12,17)(13,28,31,38,32,37,30,19)(24,39,29,35,40,33,27,36);(2,5)(6,7)(9,11)(10,17,22,34)(12,14)(13,24)(15,25)(16,26)(18,20,23,21)(19,33,38,39)(27,30)(28,36,37,35)(29,31)(32,40);(9,12,14,26,16,25,15,11)(10,18,20,17,22,23,21,34)(13,29,30,24,32,27,31,40)(19,36,28,39,38,35,37,33);(1,3,4,8)(2,6,5,7)(9,13,14,30,16,32,15,31)(10,19,20,28,22,38,21,37)(11,24,12,27,26,40,25,29)(17,33,23,36,34,39,18,35)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=f,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,d^c=di,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, 273064);