#503372

order	512
nilpotency class	4
isoclinism	isoclinic class of #501132
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×Q₈	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

742 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410568);