#136680

order	512
nilpotency class	4
isoclinism	isoclinic class of #2190
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#447
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8947
prime radical	#3414	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

353 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 353 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225462, #72225465, #72225467
maximal normal subgroup of	#72225462, #72225465, #72225467

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43876);