#133748

order	512
nilpotency class	4
isoclinism	isoclinic class of #133278
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂²×H(5,𝔽₂)	C₄
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#124 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1468 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 1468 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227927, #72227933, #72227937
maximal normal subgroup of	#72227927, #72227933, #72227937

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40944);