#151178

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

Polycyclic presentation:

2^9|c^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, 58374);