#321343

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

768 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=eg,c^2=hi,d^2=e,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, 228539);