#471454

order	512
nilpotency class	3
isoclinism	isoclinic class of #456670
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1073 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=egh,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi,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, 378650);