#197742

order	512
nilpotency class	3
isoclinism	isoclinic class of #19162
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1162 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6)(7,14)(8,12)(9,18)(10,13)(11,15)(16,19)(17,20)(21,22,26,30,27,31,25,29)(23,32,33,36,24,35,34,28);(5,7,9,15,10,16,8,17)(6,11,12,14,13,20,18,19)(21,23,27,24)(22,28,31,36)(25,33,26,34)(29,32,30,35);(2,4)(21,24,27,23)(22,28,31,36)(25,34,26,33)(29,32,30,35);(5,8,10,9)(6,12,13,18)(7,15,16,17)(11,19,20,14)(21,25,27,26)(22,29,31,30)(23,33,24,34)(28,32,36,35)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 104938);