#165903

order	512
nilpotency class	3
isoclinism	isoclinic class of #165873
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1232 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|d^2=h,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,e^b=eg,f^b=fh,e^c=eh,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257859, #72257868, #72257875
maximal normal subgroup of	#72257859, #72257868, #72257875
maximal quotient of	#72307455, #72307565

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 73099);