#510807

order	512
nilpotency class	4
isoclinism	isoclinic class of #510804
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31945
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3058	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

417 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:

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

Polycyclic presentation:

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

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, 418003);