#119993

order	512
nilpotency class	3
isoclinism	isoclinic class of #96924
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂×C₂²⋊C₄
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1090 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=dh,e^2=i,g^2=hi,b^a=bd,c^a=ce,e^a=eh,c^b=cf,e^c=ei,g^c=ghi

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, 27189);