#512154

order	512
nilpotency class	4
isoclinism	isoclinic class of #511779
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	#23902	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1106 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)(11,15)(12,17)(13,16)(14,18)(19,29)(20,27)(21,28)(22,30)(23,32)(24,37)(25,31)(26,34)(33,40)(35,39)(36,38);(2,5)(6,8)(9,11,14,19)(10,15,18,29)(12,20,23,33)(13,21,25,34)(16,28,31,26)(17,27,32,40)(22,35,24,36)(30,39,37,38);(1,3,4,7)(2,6,5,8)(10,16)(11,19)(15,26)(17,30)(18,31)(20,33)(21,34)(27,38)(28,29)(32,37)(35,36)(39,40);(9,12,14,23)(10,17,18,32)(11,20,19,33)(13,24,25,22)(15,27,29,40)(16,30,31,37)(21,36,34,35)(26,38,28,39)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419350);