#211892

order	512
nilpotency class	3
isoclinism	isoclinic class of #1685
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#121
prime radical	#3035	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

627 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)(3,7)(6,8)(9,10,13,17,14,11,18,16)(12,15,20,23,21,19,24,22);(1,2,3,5,4,6,7,8);(2,6)(5,8)(9,11)(10,14)(12,19)(13,16)(15,21)(17,18)(20,22)(23,24);(9,12,13,20,14,21,18,24)(10,15,17,23,11,19,16,22)

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=g,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 119088);