#133427

order	512
nilpotency class	4
isoclinism	isoclinic class of #133422
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

698 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267638
maximal normal subgroup of	#72267638
maximal quotient of	#72267639, #72267645

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40623);