#146013

order	512
nilpotency class	4
isoclinism	isoclinic class of #146013
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2075
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#532	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

231 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 231 conjugacy classes of subgroups — too many to draw. Only its 74 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,c^2=d,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fh,e^c=eg,f^c=fhi,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53209);