#146161

order	512
nilpotency class	4
isoclinism	isoclinic class of #144073
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

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

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

normal subgroups only

This group has 194 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) = 72
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=ei,b^2=d,c^2=dgh,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,c^b=cf,f^b=fh,h^b=hi,e^c=eg,f^c=fhi,g^c=gi,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, 53357);