#420523

order	512
nilpotency class	3
isoclinism	isoclinic class of #420522
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15278
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23412	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1620 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:

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

Polycyclic presentation:

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

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, 327719);