#508542

order	512
nilpotency class	4
isoclinism	isoclinic class of #508540
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#29925
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

451 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=i,d^2=ei,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,g^b=gh,h^b=hi,d^c=de,e^c=ei,g^c=gh,h^c=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, 415738);