#152528

order	512
nilpotency class	5
isoclinism	isoclinic class of #13669
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2184
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

324 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,d^2=gh,e^2=ghi,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=ei,d^b=dg,e^b=eg,f^b=fg,g^b=gh,h^b=hi,d^c=dg,e^c=egi,f^c=fg,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59724);