#138344

order	512
nilpotency class	4
isoclinism	isoclinic class of #138195
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

543 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 543 conjugacy classes of subgroups — too many to draw. Only its 90 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)(3,5,7,8)(4,6)(9,10,15,22,14,21,40,20,16,23,41,56,17,24,13,38)(11,25,29,18,28,47,27,46,30,60,66,48,31,54,43,49)(12,32,36,55,42,19,39,52,37,57,44,51,35,53,34,58)(26,61,64,72,67,59,62,50,33,70,68,69,63,71,65,45);(1,3,4,7)(2,5)(6,8)(9,11,17,28,16,30,14,31)(10,18,21,49,23,48,24,46)(12,33,35,63,37,26,42,67)(13,29,41,27,40,66,15,43)(19,50,57,72,53,45,32,69)(20,54,56,60,38,47,22,25)(34,68,44,65,39,64,36,62)(51,61,58,71,55,70,52,59);(3,7)(5,8)(9,12)(10,19)(11,26)(13,39)(14,42)(15,44)(16,37)(17,35)(18,45)(20,55)(21,57)(22,58)(23,53)(24,32)(25,59)(27,65)(28,67)(29,68)(30,33)(31,63)(34,40)(36,41)(38,51)(43,62)(46,72)(47,70)(48,50)(49,69)(52,56)(54,61)(60,71)(64,66)

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,c^b=cg,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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45540);