#201872

order	512
nilpotency class	3
isoclinism	isoclinic class of #201664
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#34935	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1417 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=h,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109068);