#500229

order	512
nilpotency class	4
isoclinism	isoclinic class of #499803
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	#2404	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1071 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)(11,19)(15,27)(20,31)(21,32)(28,38)(29,37)(33,39)(34,40);(2,5)(6,8)(9,10,14,18)(11,15,21,29)(12,22,24,16)(13,25,26,17)(19,27,32,37)(20,33,34,28)(23,30,36,35)(31,39,40,38);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,20)(13,19)(14,21)(16,28)(17,27)(18,29)(22,33)(23,31)(24,34)(25,37)(26,32)(30,38)(35,39)(36,40);(2,5)(6,8)(9,12,14,24)(10,16,18,22)(11,20,21,34)(13,23,26,36)(15,28,29,33)(17,30,25,35)(19,31,32,40)(27,38,37,39)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407425);