#322620

order	512
nilpotency class	3
isoclinism	isoclinic class of #298939
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

781 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)(5,6,11,20,9,14,10,15)(7,8,17,19,13,12,16,18)(21,22,26,28,25,30,24,29)(23,27,31,36,32,34,33,35);(1,2,4,3)(5,7,9,13)(6,12,14,8)(10,16,11,17)(15,19,20,18)(21,23,26,31,25,32,24,33)(22,27,28,36,30,34,29,35);(5,8,9,12)(6,13,14,7)(10,19,11,18)(15,16,20,17)(22,28)(23,31)(24,26)(27,34)(29,30)(32,33);(1,3,4,2)(6,14)(7,13)(15,20)(16,17)(23,32)(27,34)(31,33)(35,36)

Polycyclic presentation:

2^9|a^2=fh,b^2=egi,c^2=hi,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,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, 229816);