#510877

order	512
nilpotency class	4
isoclinism	isoclinic class of #509117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3199
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#32004
prime radical	#28180	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

433 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418073);