#142323

order	512
nilpotency class	4
isoclinism	isoclinic class of #142224
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

537 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 537 conjugacy classes of subgroups — too many to draw. Only its 70 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=g,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,c^b=cf,d^b=dg,h^b=hi,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49519);