#512027

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	C₂×#555	C₂²
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

908 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:

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

Polycyclic presentation:

2^9|a^2=e,b^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,c^b=ci,d^c=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, 419223);