#452327

order	512
nilpotency class	3
isoclinism	isoclinic class of #417902
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2684
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

799 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:

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

Polycyclic presentation:

2^9|a^2=egh,c^2=ef,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 359523);