#133402

order	512
nilpotency class	4
isoclinism	isoclinic class of #133394
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂²×D₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	#45442	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 871 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)(3,7,8,5)(4,6)(9,10,13,17,14,18,27,34)(11,19,22,36,20,35,38,40)(12,23,25,33,24,16,26,31)(15,28,30,21,29,39,32,37);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,21)(13,22)(14,20)(16,19)(17,32)(18,29)(23,35)(24,37)(25,39)(26,28)(27,38)(30,34)(31,40)(33,36);(11,20)(12,24)(15,29)(19,35)(22,38)(25,26)(30,32)(36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303844
maximal normal subgroup of	#72303844
maximal quotient of	#72304159, #72304167

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40598);