#118425

order	512
nilpotency class	3
isoclinism	isoclinic class of #118417
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

727 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)(5,7)(6,8)(9,10,13,17,14,12,19,16)(11,18,20,15,22,24,21,23);(1,3,5,8,6,7,4,2)(9,11,19,21,14,22,13,20)(10,15,17,24,12,23,16,18);(2,3)(4,5)(7,8)(9,12)(10,14)(11,18)(13,17)(15,21)(16,19)(20,23)(22,24)

Polycyclic presentation:

2^9|a^2=g,b^2=dh,d^2=h,e^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,f^b=fhi,d^c=di,e^c=eh,f^c=fh,g^c=ghi

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, 25621);