#93942

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	#7122
prime radical	C₂⁴	#102
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

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

normal subgroups only

This group has 211 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fhi,d^2=g,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eh,f^a=fi,c^b=cf,d^b=defh,g^b=gi,h^b=hi,d^c=dh,e^c=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, 1138);