#131405

order	512
nilpotency class	4
isoclinism	isoclinic class of #12304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1616
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 606 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)(5,6,10,16,31,45,32,15,12,17,33,46,11,18,9,30)(7,19,23,39,36,62,21,40,22,49,54,13,24,38,34,47)(8,25,28,57,59,43,26,41,27,14,35,42,29,58,60,44)(20,48,51,56,61,67,63,66,53,64,55,37,52,65,50,68);(2,4)(5,7,12,22)(6,13,17,39)(8,20,27,53)(9,23,32,54)(10,34,33,21)(11,36,31,24)(14,37,25,56)(15,38,30,62)(16,19,46,49)(18,47,45,40)(26,55,60,51)(28,50,35,63)(29,61,59,52)(41,67,44,65)(42,48,57,64)(43,68,58,66);(5,8)(6,14,17,25)(7,20)(9,26)(10,35)(11,29)(12,27)(13,37,39,56)(15,44,30,41)(16,42,46,57)(18,43,45,58)(19,48,49,64)(21,50)(22,53)(23,55)(24,52)(28,33)(31,59)(32,60)(34,63)(36,61)(38,65,62,67)(40,66,47,68)(51,54)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=hi,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=egi,f^b=fgh,h^b=hi,d^c=dg,e^c=ei,f^c=fgi,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, 38601);