#365867

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
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

796 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)(11,23,25,33,26,18,12,17)(13,28,30,36,31,35,29,19)(24,38,39,34,40,32,27,37);(2,5)(6,7)(9,11)(10,17,22,33)(12,14)(13,24)(15,25)(16,26)(18,20,23,21)(19,32,36,38)(27,29)(28,37,35,34)(30,39)(31,40);(9,12,14,26,16,25,15,11)(10,18,20,17,22,23,21,33)(13,27,29,40,31,39,30,24)(19,34,28,32,36,37,35,38);(1,3,4,8)(2,6,5,7)(9,13,14,29,16,31,15,30)(10,19,20,28,22,36,21,35)(11,24,12,27,26,40,25,39)(17,32,23,37,33,38,18,34)

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,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, 273063);