#416205

order	512
nilpotency class	3
isoclinism	isoclinic class of #400051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1326 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)(7,11,19,26)(8,21)(9,15)(10,13)(12,23)(14,16)(17,20,27,24)(18,31,32,36)(22,28)(25,34,35,33)(29,30);(1,2,3,4)(5,7,16,24,9,19,10,20)(6,11,14,17,15,26,13,27)(8,18,30,34,22,32,23,33)(12,25,21,31,29,35,28,36);(5,6)(7,17)(8,12)(9,15)(10,14)(11,24)(13,16)(18,31)(19,27)(20,26)(21,23)(22,29)(25,34)(28,30)(32,36)(33,35);(5,8,10,23,9,22,16,30)(6,12,14,21,15,29,13,28)(7,18,20,33,19,32,24,34)(11,25,17,31,26,35,27,36)

Polycyclic presentation:

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

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, 323401);