#145575

order	512
nilpotency class	4
isoclinism	isoclinic class of #144073
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1675
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#472
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

232 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52771);