#120882

order	512
nilpotency class	3
isoclinism	isoclinic class of #120880
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1442
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 359 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=h,e^2=h,f^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,d^b=di,f^b=fhi,f^c=fh,g^c=gi

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, 28078);