#504594

order	512
nilpotency class	4
isoclinism	isoclinic class of #504384
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#13316	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1753 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

534 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)(3,6)(4,5)(7,8)(10,14)(11,18)(17,21)(19,24);(2,5)(6,8)(9,10)(11,15)(12,14)(13,21)(16,19)(17,20)(18,22)(23,24);(1,3,4,7)(2,6,5,8)(10,15)(11,19)(13,20)(14,22)(16,17)(18,24)(21,23);(2,5)(6,8)(9,11,13,19,12,18,20,24)(10,15,21,16,14,22,17,23)

Polycyclic presentation:

2^9|c^2=ei,d^2=fh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,d^c=dh,g^c=gh,h^c=hi,g^d=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411790);