#301923

order	512
nilpotency class	3
isoclinism	isoclinic class of #301917
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62253	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1244 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 209119);