#502762

order	512
nilpotency class	4
isoclinism	isoclinic class of #501262
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#532	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 799 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) = 256

A minimal faithful permutation representation — 4 generators on 72 points:

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

Polycyclic presentation:

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

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, 409958);