#144687

order	512
nilpotency class	4
isoclinism	isoclinic class of #144686
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2033
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

412 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 66 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)(3,5)(4,6)(7,8)(9,10,15,20)(11,21,26,41)(12,27,31,62)(13,33,35,18)(14,36,38,67)(16,25,42,23)(17,43,46,30)(19,48,50,34)(22,53,28,64)(24,40,59,71)(29,63,32,49)(37,47,66,45)(39,57,44,69)(51,65,72,55)(52,56,68,58)(54,70,60,61);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,28)(13,25)(14,24)(15,26)(17,44)(18,41)(19,40)(20,42)(21,33)(22,31)(23,35)(27,61)(29,54)(30,65)(32,60)(34,58)(36,68)(37,69)(38,59)(39,46)(43,55)(45,51)(47,72)(48,56)(49,53)(50,71)(52,67)(57,66)(62,70)(63,64);(5,8)(9,12,13,29,15,31,35,32)(10,17,18,45,20,46,33,47)(11,22,23,54,26,28,25,60)(14,37,34,43,38,66,48,30)(16,39,21,51,42,44,41,72)(19,49,36,62,50,63,67,27)(24,57,56,55,59,69,58,65)(40,64,52,70,71,53,68,61)

Polycyclic presentation:

2^9|a^2=i,c^2=d,d^2=i,e^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fh,g^a=gi,c^b=cf,d^b=di,e^b=eh,e^c=egi,f^c=fi,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, 51883);