#146053

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

223 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 223 conjugacy classes of subgroups — too many to draw. Only its 70 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:

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

Polycyclic presentation:

2^9|a^2=eh,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, 53249);