#146630

order	512
nilpotency class	4
isoclinism	isoclinic class of #2204
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	B(4,𝔽₂)
commutator	C₂×Q₈	C₂×C₄²
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1112 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)(9,10,14,20,15,21,13,19)(11,18,23,17,12,16,22,24);(5,6,7,8)(10,16,21,18)(13,14)(17,20,24,19)(22,23);(2,4)(9,11,15,12)(10,17,21,24)(13,23,14,22)(16,20,18,19)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,e^b=ehi,f^b=fhi,d^c=dh,e^c=ei,f^c=fi,e^d=ei,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, 53826);