#273942

order	512
nilpotency class	3
isoclinism	isoclinic class of #273819
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3038	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

854 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=i,d^2=hi,e^2=i,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gh,e^c=ehi,f^c=fhi,g^c=gh,e^d=ei,f^d=fhi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 181138);