#403824

order	512
nilpotency class	3
isoclinism	isoclinic class of #395187
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	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1098 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1098 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)(9,10,14,36,16,21,15,20)(11,22,25,17,27,48,26,40)(12,28,37,18,31,42,30,43)(13,32,34,45,35,44,33,19)(23,38,56,46,51,65,50,57)(24,52,54,66,55,39,53,47)(29,59,60,63,62,41,61,58)(49,68,69,72,71,67,70,64);(1,2,4,6)(3,8,7,5)(9,11,15,26,16,27,14,25)(10,17,36,48,21,40,20,22)(12,23,37,56,31,51,30,50)(13,24,34,54,35,55,33,53)(18,38,28,57,43,65,42,46)(19,39,44,66,45,52,32,47)(29,49,61,70,62,71,60,69)(41,64,58,68,59,72,63,67);(1,3,4,7)(2,5,6,8)(9,12)(10,18)(11,23)(13,29)(14,30)(15,37)(16,31)(17,38)(19,41)(20,42)(21,43)(22,46)(24,49)(25,50)(26,56)(27,51)(28,36)(32,63)(33,60)(34,61)(35,62)(39,64)(40,65)(44,58)(45,59)(47,67)(48,57)(52,72)(53,69)(54,70)(55,71)(66,68);(9,13,14,33,16,35,15,34)(10,19,20,32,21,45,36,44)(11,24,25,53,27,55,26,54)(12,29,30,60,31,62,37,61)(17,39,22,47,40,52,48,66)(18,41,42,63,43,59,28,58)(23,49,50,69,51,71,56,70)(38,64,46,67,65,72,57,68)

Polycyclic presentation:

2^9|a^2=fgi,b^2=egi,c^2=g,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^c=fi,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, 311020);