#133378

order	512
nilpotency class	4
isoclinism	isoclinic class of #133374
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	#46427	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#537 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 851 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:

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

Polycyclic presentation:

2^9|a^2=f,e^2=i,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^c=fi,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, 40574);