#106593

order	512
nilpotency class	3
isoclinism	isoclinic class of #106570
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1139 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=hi,f^2=hi,b^a=bd,c^a=ce,d^a=dh,g^a=gh,d^b=di,e^b=eh,f^b=fi,d^c=dh

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, 13789);