#205969

order	512
nilpotency class	3
isoclinism	isoclinic class of #205777
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1060 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 113165);