#153314

order	512
nilpotency class	5
isoclinism	isoclinic class of #153083
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2215
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂²×D₃₂	C₂
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

270 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

2^9|c^2=d,d^2=gh,e^2=gi,f^2=gi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,f^a=fghi,g^a=gh,h^a=hi,c^b=cf,d^b=dg,e^b=eghi,f^b=fghi,g^b=gh,h^b=hi,e^c=ei,f^c=fi

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, 60510);