#433964

order	512
nilpotency class	3
isoclinism	isoclinic class of #418818
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15494
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

969 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:

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

Polycyclic presentation:

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

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, 341160);