#133852

order	512
nilpotency class	4
isoclinism	isoclinic class of #133825
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	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂³×D₄	C₂³
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#121 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

1119 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 1119 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2)(3,7,8,5)(4,6)(9,10)(11,17)(12,20,21,15)(13,16)(14,19,22,24)(18,23);(1,3,4,8)(2,5,6,7)(11,18)(14,22)(17,23)(19,24);(9,11,13,18)(10,14,16,22)(12,19,21,24)(15,17,20,23)

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,h^a=hi,c^b=ci,f^b=fg,f^c=fh,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260991, #72261428
maximal normal subgroup of	#72260991, #72261428

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41048);