#120873

order	512
nilpotency class	3
isoclinism	isoclinic class of #120868
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 352 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) = 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)(35,45)(36,50,54,58)(37,43)(38,42)(39,44)(40,47)(41,52,59,53)(46,61,63,55)(48,56)(49,60)(51,64,62,57);(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)(33,35,37,48,39,49,38,47)(34,40,43,45,44,56,42,60)(36,51,52,63,54,62,53,46)(41,57,50,61,59,64,58,55);(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)(33,36,37,52,39,54,38,53)(34,41,42,58,44,59,43,50)(35,46,47,62,49,63,48,51)(40,55,45,57,56,61,60,64)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28069);