#454373

order	512
nilpotency class	3
isoclinism	isoclinic class of #454372
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1835 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361569);