#138311

order	512
nilpotency class	4
isoclinism	isoclinic class of #137944
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

579 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 579 conjugacy classes of subgroups — too many to draw. Only its 86 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)(11,21)(12,27,30,46)(13,18,22,44)(14,19)(15,20)(16,24,39,31)(17,29,41,35)(23,52,55,71)(25,49)(26,50)(28,60)(32,63)(33,43,51,70)(34,65)(36,47)(37,66)(38,57,45,62)(40,64)(42,61)(48,54,67,58)(53,68)(56,59)(69,72);(2,6,8,7)(9,11,22,24,15,26,13,31)(10,16,18,21,20,39,44,50)(12,23,28,56,30,55,32,53)(14,34,51,62,36,25,33,57)(17,37,42,48,41,69,40,67)(19,45,43,65,47,38,70,49)(27,59,63,71,46,68,60,52)(29,58,61,72,35,54,64,66);(1,3,5,4)(2,6,8,7)(9,12)(10,17)(11,23)(13,32)(14,35)(15,30)(16,37)(18,42)(19,46)(20,41)(21,48)(22,28)(24,56)(25,58)(26,55)(27,47)(29,36)(31,53)(33,61)(34,54)(38,59)(39,69)(40,44)(43,60)(45,68)(49,71)(50,67)(51,64)(52,65)(57,72)(62,66)(63,70)

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=ghi,d^2=ghi,e^2=i,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,h^b=hi,d^c=di,e^c=ei,f^c=fi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45507);