#419064

order	512
nilpotency class	3
isoclinism	isoclinic class of #417661
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60371	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2235 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)(5,6,10,15,11,9,12,16)(7,13,19,28,21,18,20,29)(8,22,25,23,24,31,26,14)(17,32,34,30,33,36,35,27);(5,7)(6,13)(8,17,24,33)(9,18)(10,19)(11,21)(12,20)(14,30,23,27)(15,28)(16,29)(22,36,31,32)(25,34,26,35);(2,4)(5,8)(6,14)(7,17)(9,23)(10,26)(11,24)(12,25)(13,27)(15,31)(16,22)(18,30)(19,35)(20,34)(21,33)(28,36)(29,32);(5,9,11,6)(7,18,21,13)(8,23,24,14)(10,15,12,16)(17,30,33,27)(19,28,20,29)(22,26,31,25)(32,35,36,34)

Polycyclic presentation:

2^9|a^2=f,b^2=e,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,g^d=gi,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, 326260);