#111718

order	512
nilpotency class	3
isoclinism	isoclinic class of #111717
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1043 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^b=di,e^b=eh,g^b=ghi,e^c=eh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338220, #72338221, #72338228
maximal normal subgroup of	#72338220, #72338221, #72338228

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18914);