#512101

order	512
nilpotency class	4
isoclinism	isoclinic class of #511751
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1010 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 1010 conjugacy classes of subgroups — too many to draw. Only its 164 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419297);