#153200

order	512
nilpotency class	5
isoclinism	isoclinic class of #153199
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13348
commutator, Frattini, cosocle	#532	C₂³
prime radical	#3018	C₂²
LCS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #532 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

476 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 50 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dhi,f^a=fghi,g^a=gi,c^b=cf,d^b=dhi,e^b=ei,f^b=fi,g^b=gh,d^c=dg,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fh,g^f=gi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60396);