#153154

order	512
nilpotency class	5
isoclinism	isoclinic class of #153151
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13346
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

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

normal subgroups only

This group has 541 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60350);