#107119

order	512
nilpotency class	3
isoclinism	isoclinic class of #107114
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

1201 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gi,d^b=di,e^b=ei,f^b=fh,d^c=di

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, 14315);