#153604

order	512
nilpotency class	6
isoclinism	isoclinic class of #153601
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₃₂
commutator, Frattini, cosocle	C₂×C₃₂	C₂³
prime radical	C₂²×D₃₂	C₂
LCS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂ ◁ G
LECS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

196 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 46 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)(5,6,11,17)(7,12,22,34)(8,23,26,57)(9,27,29,60)(10,30,28,16)(13,35,38,18)(14,39,41,19)(15,42,44,24)(20,43,53,68)(21,45,52,33)(25,58,49,56)(31,62,64,47)(32,65,66,46)(36,67,63,51)(37,48,61,54)(40,50,55,59);(6,12)(7,18)(8,19)(9,24)(10,28)(13,14)(15,43)(16,45)(17,34)(20,51)(21,54)(22,35)(23,31)(25,59)(26,39)(27,32)(29,42)(30,33)(36,56)(37,55)(38,41)(40,61)(44,68)(46,47)(48,52)(49,50)(53,67)(57,64)(58,63)(60,66)(62,65);(2,4)

Polycyclic presentation:

2^9|a^2=i,d^2=fg,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,f^a=fg,g^a=gh,h^a=hi,d^b=df,f^b=fg,g^b=gh,h^b=hi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286862
maximal normal subgroup of	#72286862

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60800);