#131328

order	512
nilpotency class	4
isoclinism	isoclinic class of #12117
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1595
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

726 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 98 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)(5,6,11,17,34,46,36,15,12,18,35,47,10,16,9,32)(7,19,23,41,22,51,21,39,25,52,60,13,24,40,37,48)(8,26,29,62,28,44,27,49,31,45,64,42,30,14,33,43)(20,53,56,61,59,50,58,38,57,66,63,65,55,68,54,67);(2,4)(5,7,12,25)(6,13,18,41)(8,20,31,57)(9,23,36,60)(10,22,34,24)(11,37,35,21)(14,38,44,67)(15,40,32,51)(16,48,46,39)(17,19,47,52)(26,61,45,65)(27,63,33,56)(28,55,30,59)(29,54,64,58)(42,53,62,66)(43,68,49,50);(5,8)(6,14)(7,20)(9,33)(10,30)(11,29)(12,31)(13,38)(15,42)(16,45)(17,49)(18,44)(19,50)(21,58)(22,59)(23,56)(24,55)(25,57)(26,46)(27,36)(28,34)(32,62)(35,64)(37,54)(39,61)(40,53)(41,67)(43,47)(48,65)(51,66)(52,68)(60,63)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=hi,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dh,e^b=eg,f^b=fgh,h^b=hi,d^c=dgi,e^c=ei,f^c=fg,e^d=ei,f^d=fi,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38524);