#462454

order	512
nilpotency class	3
isoclinism	isoclinic class of #454676
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

875 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=i,c^2=i,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=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, 369650);