#404076

order	512
nilpotency class	3
isoclinism	isoclinic class of #396107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1361 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311272);