#478778

order	512
nilpotency class	3
isoclinism	isoclinic class of #476659
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

640 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)(5,6,9,14,11,16,10,15)(7,12,26,18,20,29,19,28)(8,21,24,22,23,30,25,13)(17,31,35,27,33,32,34,36);(5,7)(6,12)(8,17,23,33)(9,26)(10,19)(11,20)(13,27,22,36)(14,18)(15,28)(16,29)(21,32,30,31)(24,35,25,34);(5,8,9,25,11,23,10,24)(6,13,14,30,16,22,15,21)(7,17,19,35,20,33,26,34)(12,27,28,32,29,36,18,31);(2,4)(5,6,11,16)(7,18,20,28)(8,22,23,13)(9,15,10,14)(12,19,29,26)(17,32,33,31)(21,25,30,24)(27,34,36,35)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385974);