#264967

order	512
nilpotency class	3
isoclinism	isoclinic class of #264939
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×D₄×D₈	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1265 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|b^2=eh,c^2=i,d^2=eh,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,f^b=fhi,g^b=ghi,f^c=fi,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 172163);