#140557

order	512
nilpotency class	4
isoclinism	isoclinic class of #13215
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1961
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#418
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

377 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 377 conjugacy classes of subgroups — too many to draw. Only its 102 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=eg,d^2=gi,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=cg,e^b=ei,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47753);