#513104

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

Polycyclic presentation:

2^9|a^2=i,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, 420300);