#235350

order	512
nilpotency class	3
isoclinism	isoclinic class of #235342
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60561	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1406 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142546);