#499219

order	512
nilpotency class	4
isoclinism	isoclinic class of #498886
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

965 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=e,d^2=i,b^a=be,c^a=cf,d^a=di,g^a=ghi,c^b=cg,f^b=fh,h^c=hi,g^f=gi

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, 406415);