#452984

order	512
nilpotency class	3
isoclinism	isoclinic class of #421163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

908 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)(3,6)(4,5)(7,8)(9,10,14,20,16,21,15,19)(11,22,24,17,12,26,25,34)(13,28,31,18,32,36,30,37)(23,39,40,38,27,33,29,35);(2,5)(3,7)(9,11,14,24,16,12,15,25)(10,17,19,22,21,34,20,26)(13,29,30,27,32,40,31,23)(18,35,36,39,37,38,28,33);(9,12,16,11)(10,17,21,34)(13,27,32,23)(14,24,15,25)(18,33,37,39)(19,26,20,22)(28,35,36,38)(29,31,40,30);(1,3,4,7)(2,6,5,8)(9,13,16,32)(10,18)(11,23,12,27)(14,31,15,30)(17,33)(19,28)(20,36)(21,37)(22,38)(24,40,25,29)(26,35)(34,39)

Polycyclic presentation:

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

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, 360180);