#142348

order	512
nilpotency class	4
isoclinism	isoclinic class of #142236
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#29738	C₂
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 493 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) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 52 points:

(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,19)(18,20)(21,22)(23,29)(24,28)(25,31)(26,40)(27,32)(30,36)(33,45)(34,41)(35,47)(37,42)(38,43)(39,46)(44,48)(49,52)(50,51);(5,7,9,14)(6,10,12,17)(8,13,15,20)(11,18,19,16)(22,28)(23,33)(24,36)(25,26)(29,44)(30,31)(32,43)(34,49)(35,39)(37,38)(40,42)(41,45)(46,52)(47,51);(2,4)(5,8)(6,11)(7,13)(9,15)(10,16)(12,19)(14,20)(17,18)(21,23)(22,29)(24,34)(25,39)(26,33)(27,35)(28,41)(30,44)(31,46)(32,47)(36,48)(37,50)(38,49)(40,45)(42,51)(43,52)

Polycyclic presentation:

2^9|b^2=h,d^2=gi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fh,g^a=gi,c^b=cf,d^b=dg,f^b=fi,g^b=gi,d^c=dh,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49544);