#474312

order	512
nilpotency class	3
isoclinism	isoclinic class of #458126
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2495
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

627 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381508);