#232129

order	512
nilpotency class	3
isoclinism	isoclinic class of #232128
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

622 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=efh,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=eh,c^b=cefi,d^b=de,f^b=fh,g^b=gh,g^c=gh

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, 139325);