#153172

order	512
nilpotency class	5
isoclinism	isoclinic class of #153171
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	#33550	C₂
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

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

normal subgroups only

This group has 552 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=hi,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dhi,e^a=ei,f^a=fgh,g^a=gi,c^b=cf,d^b=dhi,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

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60368);