#152174

order	512
nilpotency class	5
isoclinism	isoclinic class of #13669
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂×#3415	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

638 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 106 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)(5,6,9,16,12,18,11,36)(7,19,21,46,23,45,31,14)(8,24,27,55,30,57,29,38)(10,33,32,52,35,66,26,17)(13,37,39,53,42,59,41,20)(15,47,49,62,50,61,56,28)(22,44,34,48,54,68,64,65)(25,51,58,40,60,67,63,43);(5,7)(6,13)(8,25)(9,21)(10,34)(11,31)(12,23)(14,38)(15,48)(16,39)(17,51)(18,42)(19,24)(20,28)(22,26)(27,58)(29,63)(30,60)(32,54)(33,40)(35,64)(36,41)(37,47)(43,66)(44,56)(45,57)(46,55)(49,68)(50,65)(52,67)(53,62)(59,61);(2,4)(5,7)(6,14)(8,20)(9,31)(10,22)(11,21)(12,23)(13,38)(15,43)(16,45)(17,44)(18,46)(19,36)(24,41)(25,28)(26,34)(27,59)(29,37)(30,53)(32,64)(33,65)(35,54)(39,57)(40,50)(42,55)(47,63)(48,66)(49,67)(51,56)(52,68)(58,61)(60,62)

Polycyclic presentation:

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

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, 59370);