#513103

order	512
nilpotency class	5
isoclinism	isoclinic class of #513098
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

375 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 375 conjugacy classes of subgroups — too many to draw. Only its 110 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)(4,6)(5,8)(9,10)(11,24)(12,23)(13,20)(14,34)(15,39)(16,19)(17,26)(18,27)(21,37)(22,36)(25,42)(28,60)(29,48)(30,63)(31,52)(32,41)(33,50)(35,55)(38,68)(40,53)(43,66)(44,62)(45,61)(46,58)(47,64)(49,69)(51,67)(54,65)(56,57)(59,71)(70,72);(1,3)(2,5)(4,8)(6,7)(9,11,13,26)(10,17,20,24)(12,29,16,25)(14,27,32,58)(15,28,33,59)(18,34,46,41)(19,48,23,42)(21,43,51,55)(22,44,52,53)(30,61,38,57)(31,62,36,40)(35,67,66,37)(39,71,50,60)(45,63,56,68)(47,49,72,54)(64,65,70,69);(3,8)(5,7)(9,11)(10,18)(12,25)(13,26)(14,35)(15,40)(16,29)(17,41)(19,45)(20,46)(21,53)(22,55)(23,56)(24,34)(27,37)(28,36)(30,64)(31,59)(32,66)(33,62)(38,70)(39,47)(42,63)(43,52)(44,51)(48,68)(49,60)(50,72)(54,71)(57,65)(58,67)(61,69);(2,6)(5,7)(9,12,13,16)(10,19,20,23)(11,25,26,29)(14,30,32,38)(15,31,33,36)(17,42,24,48)(18,45,46,56)(21,49,51,54)(22,50,52,39)(27,57,58,61)(28,40,59,62)(34,68,41,63)(35,64,66,70)(37,65,67,69)(43,47,55,72)(44,71,53,60)

Polycyclic presentation:

2^9|b^2=i,d^2=i,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fg,g^a=gh,h^a=hi,c^b=ce,d^b=di,f^c=fg,g^c=gh,h^c=hi

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, 420299);