#488968

order	512
nilpotency class	4
isoclinism	isoclinic class of #488912
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	C₄×C₈	C₂⁴
prime radical	C₂×#3273	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

987 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 987 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fhi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fhi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396164);