#503749

order	512
nilpotency class	4
isoclinism	isoclinic class of #503748
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#26044
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

478 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410945);