#146458

order	512
nilpotency class	4
isoclinism	isoclinic class of #11499
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂³×D₄	C₂³
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1964 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,f^b=fh,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53654);