#403821

order	512
nilpotency class	3
isoclinism	isoclinic class of #395181
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1114 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1114 conjugacy classes of subgroups — too many to draw. Only its 210 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 — 4 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=egi,c^2=g,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,h^c=hi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311017);