#121216

order	512
nilpotency class	3
isoclinism	isoclinic class of #121215
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1443
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

338 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 106 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)(3,13,16,24,17,28,15,8)(4,18,20,9,22,26,21,27)(14,30,19,29,32,23,31,25)(33,34)(35,44,48,59)(36,49)(37,42)(38,43)(39,54)(40,46,56,47)(41,52)(45,57,61,62)(50,55,60,64)(51,63)(53,58);(1,3)(2,8,12,24)(4,19)(5,15)(6,16)(7,17)(9,25,27,29)(10,13,11,28)(14,21)(18,23,26,30)(20,32)(22,31)(33,35,39,46,38,48,37,47)(34,40,54,44,43,56,42,59)(36,50,53,45,52,60,51,61)(41,57,63,64,49,62,58,55);(1,4,5,20,7,22,6,21)(2,9,10,18,12,27,11,26)(3,14,15,19,17,32,16,31)(8,23,13,29,24,30,28,25)(33,36,37,51,38,52,39,53)(34,41,42,58,43,49,54,63)(35,45,46,60,48,61,47,50)(40,55,44,57,56,64,59,62)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=hi,e^2=h,f^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fhi,g^b=gi,f^c=fh,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, 28412);