#121932

order	512
nilpotency class	3
isoclinism	isoclinic class of #121929
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

354 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 112 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)(36,47,49,40)(37,50)(38,42)(39,45)(41,52)(44,58,60,64)(46,57)(48,56,63,61)(51,54)(53,62,55,59);(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)(33,35,37,51,38,46,52,45)(34,39,50,43,42,54,41,57)(36,44,48,62,49,60,63,59)(40,55,61,64,47,53,56,58);(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)(33,36,37,48,38,49,52,63)(34,40,50,61,42,47,41,56)(35,44,51,62,46,60,45,59)(39,53,43,58,54,55,57,64)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29128);