#121767

order	512
nilpotency class	3
isoclinism	isoclinic class of #121766
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1445
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 366 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)(3,13,16,24,17,28,15,8)(4,18,20,9,22,26,21,27)(14,30,19,29,32,23,31,25)(33,34)(35,43,46,56)(36,40)(37,41)(38,42)(39,45,52,49)(44,59,61,58)(47,54)(48,55)(50,57)(51,62,53,60)(63,64);(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)(33,35,37,49,38,46,50,45)(34,39,41,56,42,52,57,43)(36,44,47,62,48,61,63,60)(40,53,54,58,55,51,64,59);(1,4)(2,9,12,27)(3,14)(5,20)(6,21)(7,22)(8,23,24,30)(10,18,11,26)(13,29,28,25)(15,19)(16,31)(17,32)(33,36,37,47,38,48,50,63)(34,40,41,54,42,55,57,64)(35,44,49,62,46,61,45,60)(39,51,56,59,52,53,43,58)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=hi,e^2=h,f^2=h,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=fh,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, 28963);