#198942

order	512
nilpotency class	3
isoclinism	isoclinic class of #198789
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1338 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338213, #72338256
maximal normal subgroup of	#72338213, #72338256

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106138);