#200533

order	512
nilpotency class	3
isoclinism	isoclinic class of #200400
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1182 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=i,d^2=i,e^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,e^b=ei,f^b=fi,g^b=gi,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, 107729);