#459755

order	512
nilpotency class	3
isoclinism	isoclinic class of #455570
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

928 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=i,d^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi,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, 366951);