#254312

order	512
nilpotency class	3
isoclinism	isoclinic class of #254299
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#24260	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1063 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fi,g^b=ghi,d^c=di,e^c=eh,f^c=fi,g^c=gi,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161508);