#138279

order	512
nilpotency class	4
isoclinism	isoclinic class of #137944
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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	#120
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 723 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)(11,21)(12,26,29,46)(13,30,34,18)(14,19)(15,20)(16,23,40,31)(17,28,42,36)(22,51,54,71)(24,49)(25,50)(27,61)(32,60)(33,63,57,44)(35,66)(37,47)(38,67)(39,56,45,64)(41,62)(43,65)(48,53,68,58)(52,69)(55,59)(70,72);(2,6,8,7)(9,11,13,31)(10,16,30,50)(12,22,32,52)(14,35,33,56)(15,25,34,23)(17,38,41,68)(18,21,20,40)(19,45,63,49)(24,57,64,37)(26,59,60,51)(27,55,29,54)(28,58,65,67)(36,53,62,72)(39,44,66,47)(42,70,43,48)(46,69,61,71);(9,12,15,29)(10,17,20,42)(11,22,25,54)(13,32,34,27)(14,36,37,28)(16,38,40,70)(18,43,30,41)(19,46,47,26)(21,48,50,68)(23,55,31,52)(24,58,35,53)(33,62,57,65)(39,59,45,69)(44,60,63,61)(49,71,66,51)(56,72,64,67)

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,c^2=i,d^2=ghi,e^2=i,f^2=gh,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, 45475);