#131893

order	512
nilpotency class	4
isoclinism	isoclinic class of #131820
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

599 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 599 conjugacy classes of subgroups — too many to draw. Only its 106 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=hi,e^2=g,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=egi,f^b=fgh,h^b=hi,d^c=dgi,e^c=eg

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, 39089);