#488918

order	512
nilpotency class	4
isoclinism	isoclinic class of #488916
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1011 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 1011 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) = 128

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

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

Polycyclic presentation:

2^9|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, 396114);