#510117

order	512
nilpotency class	4
isoclinism	isoclinic class of #510115
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31314
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#25803	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

562 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:

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

Polycyclic presentation:

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

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, 417313);