#134028

order	512
nilpotency class	4
isoclinism	isoclinic class of #133883
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	#35329	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#535 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

1099 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 1099 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=hi,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,f^b=fgi,e^c=ei,f^c=fhi,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41224);