#121852

order	512
nilpotency class	3
isoclinism	isoclinic class of #121851
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1445
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

373 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 373 conjugacy classes of subgroups — too many to draw. Only its 116 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=hi,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=di,e^b=ei,f^b=fi,g^b=gi,d^c=dh,e^c=eh,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29048);