#105930

order	512
nilpotency class	3
isoclinism	isoclinic class of #105909
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	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1009 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=i,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,c^b=ch,d^b=di,e^b=ei,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13126);