#133985

order	512
nilpotency class	4
isoclinism	isoclinic class of #133922
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, prime radical, cosocle	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2308 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

786 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 82 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)(3,7,4,6)(5,8)(9,10)(11,19)(12,23)(13,26,28,17)(14,18)(15,21,30,38)(16,25)(20,33)(22,34)(24,31,27,32)(29,37,39,35)(36,40);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,29)(17,19)(18,30)(23,39)(24,35)(25,36)(26,34)(27,37)(28,38)(31,40)(32,33);(1,4,5,3)(2,7,8,6)(9,12,14,25)(10,16)(11,20,22,36)(13,27,28,24)(15,29)(17,32)(18,23)(19,33)(21,37,38,35)(26,31)(30,39)(34,40)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,e^b=ei,f^b=fg,d^c=di,e^c=ei,f^c=fh,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, 41181);