#119666

order	512
nilpotency class	3
isoclinism	isoclinic class of #119665
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

787 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|a^2=g,c^2=di,d^2=h,e^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=dh,f^b=fh,e^c=ei,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26862);