#93816

order	512
nilpotency class	5
isoclinism	isoclinic class of #93814
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7080
commutator	C₂²×C₁₆	C₂×C₄
Frattini, cosocle	#2151	C₂²
prime radical	C₂²×D₁₆	C₄
LCS	G ▷ C₂²×C₁₆ ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2151 ◁ G
LECS	G ▷ #2151 ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

135 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 135 conjugacy classes of subgroups — too many to draw. Only its 26 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:

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

Polycyclic presentation:

2^9|a^2=d,c^2=fh,d^2=gi,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fh,h^a=hi,c^b=cf,d^b=defh,e^b=eh,f^b=fh,g^b=gi,h^b=hi,d^c=dghi,e^d=ei

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1012);