#106051

order	512
nilpotency class	3
isoclinism	isoclinic class of #106040
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

706 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 182 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=h,e^2=i,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=ghi,d^b=di,e^b=ei,d^c=di,e^c=ei,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13247);