#187677

order	512
nilpotency class	3
isoclinism	isoclinic class of #187676
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	#62834	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2457 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=i,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,f^c=fh

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, 94873);