#489014

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	#19133	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

728 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

2^9|b^2=hi,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, 396210);