#415122

order	512
nilpotency class	3
isoclinism	isoclinic class of #402003
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2494
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15183
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

835 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:

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

Polycyclic presentation:

2^9|a^2=fh,c^2=e,d^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^c=fi,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, 322318);