#152642

order	512
nilpotency class	5
isoclinism	isoclinic class of #13700
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2215
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#41	D₈
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

302 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=ghi,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=di,e^b=ei,f^b=fi,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59838);