#146028

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	C₂×#135	D₄
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

229 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 229 conjugacy classes of subgroups — too many to draw. Only its 74 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)(3,6)(4,5)(7,8)(9,10,18,43,13,21,40,55,17,25,14,22,16,24,15,23)(11,26,39,68,38,19,41,54,30,53,29,56,28,47,67,46)(12,31,42,51,33,65,35,20,37,57,34,49,36,50,44,52)(27,59,32,45,63,48,64,58,62,72,61,71,60,70,66,69);(2,5)(6,8)(9,11)(10,19)(12,32)(13,38)(14,39)(15,41)(16,28)(17,30)(18,29)(20,48)(21,53)(22,54)(23,56)(24,26)(25,47)(27,35)(31,58)(33,64)(34,63)(36,66)(37,61)(40,67)(42,60)(43,46)(44,62)(45,50)(49,72)(51,59)(52,70)(55,68)(57,69)(65,71);(1,3,4,7)(2,6,5,8)(9,12)(10,20)(11,27,30,62)(13,36)(14,35)(15,42)(16,33)(17,37)(18,44)(19,45,47,71)(21,51)(22,50)(23,57)(24,49)(25,52)(26,58,53,69)(28,63,38,60)(29,64,39,66)(31,55)(32,41,61,67)(34,40)(43,65)(46,72,54,59)(48,56,70,68)

Polycyclic presentation:

2^9|a^2=ei,c^2=dg,d^2=i,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=di,e^b=ei,f^b=fh,h^b=hi,d^c=di,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, 53224);