#133917

order	512
nilpotency class	4
isoclinism	isoclinic class of #133916
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂³×D₄	C₂³
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2308 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

985 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 985 conjugacy classes of subgroups — too many to draw. Only its 86 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)(11,19)(12,24)(13,27,28,17)(14,18)(15,21,31,22)(16,25,32,40)(20,34)(23,30)(26,39)(29,37)(33,36,38,35);(1,3,4,8)(2,5,6,7)(9,11)(10,15,18,31)(12,20)(13,22)(14,23)(16,29,32,34)(17,30,27,19)(21,28)(24,38,39,33)(25,36)(26,37)(35,40);(9,12,14,26)(10,16,18,32)(11,20,23,37)(13,25,28,40)(15,29,31,34)(17,24,27,39)(19,33,30,38)(21,35,22,36)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,d^b=di,f^b=fg,f^c=fh,f^d=fi,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303829, #72303830, #72303838, …
maximal normal subgroup of	#72303829, #72303830, #72303838, …
maximal quotient of	#72303913

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41113);