#465451

order	512
nilpotency class	3
isoclinism	isoclinic class of #457090
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
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

794 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=f,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^c=de,e^c=ei,h^c=hi,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, 372647);