#151179

order	512
nilpotency class	5
isoclinism	isoclinic class of #151176
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2184
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	#33765	C₂
LCS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ #2113 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

291 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 291 conjugacy classes of subgroups — too many to draw. Only its 44 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(5,8)(9,10,15,22)(11,24,28,51)(12,17,33,42)(13,23,16,20)(14,37,38,52)(18,35,46,26)(19,47,49,29)(21,36,53,31)(25,54,30,65)(27,60,61,45)(32,43,34,40)(39,59,68,67)(41,63,50,64)(44,69,48,70)(55,62,72,57)(56,66,71,58);(1,2)(3,5)(4,6)(7,8)(10,17)(12,29)(13,16)(14,31)(18,39)(19,21)(20,43)(22,42)(23,40)(24,54)(25,57)(26,35)(27,58)(30,62)(32,64)(33,47)(34,63)(36,38)(37,41)(44,60)(45,48)(46,68)(49,53)(50,52)(51,65)(55,56)(59,69)(61,66)(67,70)(71,72);(1,3)(2,5)(4,7)(6,8)(9,11)(10,18)(12,30)(13,35)(14,27)(15,28)(16,26)(17,39)(19,48)(20,51)(21,45)(22,46)(23,24)(25,33)(29,62)(31,58)(32,59)(34,67)(36,66)(37,56)(38,61)(40,54)(41,55)(42,68)(43,65)(44,49)(47,57)(50,72)(52,71)(53,60)(63,70)(64,69)

Polycyclic presentation:

2^9|a^2=i,d^2=fh,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=ei,f^a=fh,h^a=hi,d^b=df,e^b=eg,f^b=fh,h^b=hi,d^c=dg,e^c=ei

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, 58375);