#120935

order	512
nilpotency class	3
isoclinism	isoclinic class of #120932
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	#125
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 351 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:

(1,2,6,12,7,10,5,11)(3,13,15,28,16,8,22,24)(4,17,20,27,19,9,21,26)(14,23,31,32,30,29,18,25)(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)(2,8)(4,18,19,31)(5,15)(6,22)(7,16)(9,25,17,32)(10,13)(11,24)(12,28)(14,20,30,21)(23,26,29,27)(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)(2,9,10,17)(3,14)(5,21)(6,20)(7,19)(8,23,13,29)(11,27,12,26)(15,31)(16,30)(18,22)(24,32,28,25)(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=i,e^2=i,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,f^b=fi,g^b=ghi,e^c=ehi,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, 28131);