#321386

order	512
nilpotency class	3
isoclinism	isoclinic class of #298711
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

791 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eghi,c^2=hi,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,f^c=fh,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, 228582);