#137775

order	512
nilpotency class	4
isoclinism	isoclinic class of #137768
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

589 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 589 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=i,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,d^b=dgi,e^b=ei,f^b=fgi,g^b=gi,d^c=di,e^c=ei,f^c=fi,f^d=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, 44971);