#136599

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	#8939
prime radical	#3054	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

415 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 415 conjugacy classes of subgroups — too many to draw. Only its 114 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)(3,13,18,40,17,41,19,35)(4,20,22,9)(5,10)(6,12)(7,11)(8,16,30,49,32,48,31,24)(14,28)(15,39,46,64,45,59,50,62)(21,36,25,33)(23,52,54,34)(26,37)(27,38)(29,44,60,53,51,47,61,56)(42,63,55,57)(43,58);(1,3,14,16,6,18,27,49,7,17,43,48,26,19,5,24)(2,8,28,13,12,30,38,40,11,32,58,41,37,31,10,35)(4,15,42,47,23,50,21,53,22,45,55,44,54,46,25,56)(9,29,57,39,34,61,33,62,20,51,63,59,52,60,36,64);(1,4)(2,9)(3,15)(5,25)(6,23)(7,22)(8,29)(10,36)(11,20)(12,34)(13,39)(14,42)(16,47)(17,45)(18,50)(19,46)(21,27)(24,56)(26,54)(28,57)(30,61)(31,60)(32,51)(33,38)(35,64)(37,52)(40,62)(41,59)(43,55)(44,48)(49,53)(58,63)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,e^a=ei,d^b=dg,e^b=ei,f^b=fg,g^b=gi,h^b=hi,d^c=di,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43795);