#93987

order	512
nilpotency class	5
isoclinism	isoclinic class of #7440
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1353
commutator	C₂×M₁₆	C₂×C₈
Frattini, cosocle	#1729	C₂²
socle	C₂	#7132
prime radical	C₂⁴	#103
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #1729 ◁ G
LECS	G ▷ #1729 ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁

203 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 203 conjugacy classes of subgroups — too many to draw. Only its 46 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 — 2 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=fh,c^2=f,d^2=g,e^2=i,f^2=i,b^a=bc,c^a=ce,e^a=eh,h^a=hi,c^b=cf,d^b=def,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=dhi,e^c=ei,e^d=ei

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, 1183);