#352950

order	512
nilpotency class	3
isoclinism	isoclinic class of #352047
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1012 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:

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

Polycyclic presentation:

2^9|b^2=eh,c^2=f,d^2=gi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,d^c=dh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260146);