#122034

order	512
nilpotency class	3
isoclinism	isoclinic class of #97012
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

568 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fh,d^2=i,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=di,e^b=ehi,g^b=ghi,d^c=dhi,e^c=eh

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, 29230);