#120984

order	512
nilpotency class	3
isoclinism	isoclinic class of #120982
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

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=h,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,d^b=di,f^b=fhi,e^c=ehi,f^c=fi,g^c=gh

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, 28180);