#145028

order	512
nilpotency class	4
isoclinism	isoclinic class of #145027
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#532	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 376 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=dg,d^2=i,e^2=gi,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fhi,g^a=gi,c^b=cf,d^b=di,e^b=eh,f^b=fhi,d^c=di,e^c=eg,f^c=fh,g^c=gi

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, 52224);