#511584

order	512
nilpotency class	4
isoclinism	isoclinic class of #511583
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32707
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#32951	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3573 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

593 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418780);