#273930

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	C₂×AGL(1,ℤ/8)	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

841 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)(9,10,15,19)(11,20,17,24)(12,16,22,21)(13,23,14,18);(1,3,5,2)(4,7,6,8)(9,11,15,17)(10,12,19,22)(13,16,14,21)(18,24,23,20);(1,2,5,3)(4,8,6,7)(9,12)(10,16)(11,18)(13,20)(14,24)(15,22)(17,23)(19,21);(1,3,5,2)(4,7,6,8)(9,12)(10,17)(11,19)(13,24)(14,20)(15,22)(16,23)(18,21)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=hi,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, 181126);