#263222

order	512
nilpotency class	3
isoclinism	isoclinic class of #263219
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1105 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 1105 conjugacy classes of subgroups — too many to draw. Only its 164 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 — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|b^2=eh,d^2=eh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=di,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265388, #72265543
maximal normal subgroup of	#72265388, #72265543

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170418);