#488975

order	512
nilpotency class	4
isoclinism	isoclinic class of #488916
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	#61086	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 894 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=i,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fhi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,d^c=di,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, 396171);