#435785

order	512
nilpotency class	3
isoclinism	isoclinic class of #420055
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2476
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15538
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

988 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=f,e^2=i,g^2=i,h^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,g^b=gi,e^c=ei,g^c=gi,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, 342981);