#129274

order	512
nilpotency class	4
isoclinism	isoclinic class of #12711
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1781
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8299
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

797 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 797 conjugacy classes of subgroups — too many to draw. Only its 132 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36470);