#512139

order	512
nilpotency class	4
isoclinism	isoclinic class of #511802
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	#23917	C₂
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1023 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)(12,16)(13,17)(14,20)(18,24)(19,26)(21,31)(22,28)(23,34)(25,32)(27,33)(29,36)(30,35)(37,40)(38,39);(2,5)(6,8)(9,11)(10,15)(12,22)(13,23)(14,21)(16,28)(17,34)(18,30)(19,29)(20,31)(24,35)(25,37)(26,36)(27,38)(32,40)(33,39);(1,3,4,7)(2,6,5,8)(10,16)(12,24)(13,26)(15,28)(17,32)(18,20)(19,33)(22,35)(23,36)(29,39)(30,31)(34,40);(10,17)(11,21)(13,26)(15,29)(16,32)(18,33)(19,20)(22,35)(25,27)(28,39)(30,40)(31,34)

Polycyclic presentation:

2^9|c^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,g^a=gi,h^a=hi,d^b=di,f^c=fi,g^c=ghi,f^d=fh,g^d=gi,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, 419335);