#136683

order	512
nilpotency class	4
isoclinism	isoclinic class of #13033
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1879
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8947
prime radical	C₂×#508	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 351 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)(3,14,18,47,20,36,19,42)(4,21,24,57,25,58,33,43)(5,26)(6,11)(7,12)(8,13)(9,17,37,30,39,54,38,27)(10,23,40,28,41,22,44,31)(15,48)(16,49,51,35)(29,45)(32,34)(46,59,56,53)(50,61,60,62)(52,64,55,63);(1,3,15,17,29,19,5,27,8,20,32,54,7,18,6,30)(2,9,34,14,45,38,11,42,13,39,48,36,12,37,26,47)(4,16,23,53,24,55,28,60,25,51,22,59,33,52,31,50)(10,35,58,46,40,64,43,62,41,49,21,56,44,63,57,61);(1,4)(2,10)(3,16)(5,28)(6,31)(7,33)(8,25)(9,35)(11,43)(12,44)(13,41)(14,46)(15,23)(17,53)(18,52)(19,55)(20,51)(21,48)(22,32)(24,29)(26,57)(27,60)(30,50)(34,58)(36,56)(37,63)(38,64)(39,49)(40,45)(42,62)(47,61)(54,59)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=g,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=di,d^b=dg,e^b=egi,f^b=fg,g^b=gi,h^b=hi,d^c=dgi,e^c=egi,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	#72225428, #72225466, #72225467
maximal normal subgroup of	#72225428, #72225466, #72225467

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43879);