#138252

order	512
nilpotency class	4
isoclinism	isoclinic class of #138132
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 707 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,d^2=ghi,e^2=i,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=ei,f^b=fh,h^b=hi,d^c=di,e^c=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, 45448);