#450128

order	512
nilpotency class	3
isoclinism	isoclinic class of #418535
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1011 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 357324);