#151510

order	512
nilpotency class	5
isoclinism	isoclinic class of #151509
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12291
commutator, Frattini, cosocle	C₂×M₃₂	C₂³
prime radical	C₂³	#509
LCS	G ▷ C₂×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2150 ◁ G
LECS	G ▷ C₂×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

192 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 42 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,d^2=fhi,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=ei,f^a=fh,g^a=gi,h^a=hi,d^b=dfi,e^b=egi,f^b=fh,h^b=hi,d^c=dg,e^d=ei

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320875, #72342972, #72342977
maximal normal subgroup of	#72320875, #72342972, #72342977

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58706);