#121961

order	512
nilpotency class	3
isoclinism	isoclinic class of #121960
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

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

normal subgroups only

This group has 372 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) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

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

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, 29157);