#505655

order	512
nilpotency class	4
isoclinism	isoclinic class of #500592
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3253
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#27626
prime radical	#3050	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

453 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 453 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=e,e^2=i,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^c=fhi,g^c=gi,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, 412851);