#200972

order	512
nilpotency class	3
isoclinism	isoclinic class of #200914
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3151	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1530 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,d^2=h,e^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fh,g^c=gi

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, 108168);