#428807

order	512
nilpotency class	3
isoclinism	isoclinic class of #428806
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15390
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1012 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 336003);