#142283

order	512
nilpotency class	4
isoclinism	isoclinic class of #142252
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#135
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

551 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 551 conjugacy classes of subgroups — too many to draw. Only its 72 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)(11,21,25,51)(12,26,29,17)(13,31,35,49)(14,36,33,19)(16,38,41,23)(18,45,47,34)(22,52,27,50)(24,55,37,46)(28,60,63,65)(30,44,62,57)(32,48,67,40)(39,53,42,58)(43,64,56,71)(54,72,59,70)(61,69,66,68);(5,8)(9,11,15,25)(10,16,20,41)(12,27,29,22)(13,32,35,67)(14,37,33,24)(17,42,26,39)(18,46,47,55)(19,48,36,40)(21,49,51,31)(23,45,38,34)(28,61,63,66)(30,54,62,59)(43,70,56,72)(44,68,57,69)(50,65,52,60)(53,64,58,71);(1,3)(2,5)(4,7)(6,8)(9,12,14,30,15,29,33,62)(10,17,19,44,20,26,36,57)(11,22,24,54,25,27,37,59)(13,28,34,64,35,63,45,71)(16,39,40,68,41,42,48,69)(18,43,31,65,47,56,49,60)(21,50,46,72,51,52,55,70)(23,53,32,66,38,58,67,61)

Polycyclic presentation:

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

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, 49479);