#133960

order	512
nilpotency class	4
isoclinism	isoclinic class of #133875
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂³×D₄	C₂³
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#127 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

1330 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 1330 conjugacy classes of subgroups — too many to draw. Only its 94 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)(3,7,4,8)(5,6)(9,10)(11,17,19,14)(12,15)(13,16)(18,23,24,21)(20,22);(1,3,5,4)(2,6)(7,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23);(2,7,6,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=g,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,f^b=fgi,f^c=fhi,f^d=fi,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232994, #72260719
maximal normal subgroup of	#72232994, #72260719
maximal quotient of	#72232982, #72261479

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41156);