#499944

order	512
nilpotency class	4
isoclinism	isoclinic class of #499919
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60685	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

939 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 939 conjugacy classes of subgroups — too many to draw. Only its 152 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fi,f^c=fh,h^c=hi

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, 407140);