#107319

order	512
nilpotency class	3
isoclinism	isoclinic class of #107314
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	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ 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:

(1,2,5,7,6,3,4,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,21)(20,22)(23,24);(1,3,6,2)(4,7,5,8)(9,11)(10,14,16,17)(12,18)(13,19)(15,21,22,23)(20,24);(1,2,6,3)(4,7,5,8)(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,c^2=h,d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,d^b=di,e^b=ehi,d^c=dhi,e^c=eh,f^c=fh

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, 14515);