#385803

order	512
nilpotency class	3
isoclinism	isoclinic class of #31414
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2688
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14415
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

938 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,7,12,8,13,6,31)(3,14,17,9,19,34,18,35)(4,20,23,38,24,10,25,39)(5,26,29,42,30,41,28,11)(15,36,46,60,45,43,21,32)(16,47,27,40,48,33,49,61)(22,52,54,37,55,57,56,50)(44,59,58,62,53,63,51,64);(1,3)(2,9)(4,21,24,46)(5,27)(6,18)(7,17)(8,19)(10,36,20,43)(11,40)(12,34)(13,35)(14,31)(15,25,45,23)(16,28)(22,53,55,44)(26,33)(29,48)(30,49)(32,39,60,38)(37,63,50,59)(41,47)(42,61)(51,56,58,54)(52,62,57,64);(1,4,6,25,8,24,7,23)(2,10,12,39,13,20,31,38)(3,15,17,46,19,45,18,21)(5,22,28,56,30,55,29,54)(9,32,14,43,35,60,34,36)(11,37,26,57,42,50,41,52)(16,44,27,58,48,53,49,51)(33,59,40,64,47,63,61,62);(1,5,6,28,8,30,7,29)(2,11,12,26,13,42,31,41)(3,16,17,27,19,48,18,49)(4,22,25,56,24,55,23,54)(9,33,14,40,35,47,34,61)(10,37,39,57,20,50,38,52)(15,44,46,58,45,53,21,51)(32,59,43,64,60,63,36,62)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=ef,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,e^c=ei,f^c=fi,g^d=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, 292999);