#509342

order	512
nilpotency class	4
isoclinism	isoclinic class of #509340
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#30593
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3067	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

524 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:

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

Polycyclic presentation:

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

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, 416538);