#433017

order	512
nilpotency class	3
isoclinism	isoclinic class of #419839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

965 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,c^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,g^c=gi,h^c=hi,f^d=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, 340213);