#186217

order	512
nilpotency class	3
isoclinism	isoclinic class of #186146
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62800	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2432 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=hi,e^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dhi,e^a=eh,e^b=ei,g^b=ghi,f^c=fh,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93413);