#153383

order	512
nilpotency class	5
isoclinism	isoclinic class of #153105
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2206
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	#509
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

174 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=d,c^2=d,e^2=i,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,c^b=cf,f^b=fg,g^b=gh,h^b=hi,e^c=ei,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, 60579);