#422683

order	512
nilpotency class	3
isoclinism	isoclinic class of #419041
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1049 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=ei,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,f^c=fi,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, 329879);