#94659

order	512
nilpotency class	6
isoclinism	isoclinic class of #94658
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7408
commutator	#423	C₂×C₄
Frattini, cosocle	#1963	C₂²
prime radical	#532	D₄
LCS	G ▷ #423 ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1963 ◁ G
LECS	G ▷ #1963 ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

171 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 171 conjugacy classes of subgroups — too many to draw. Only its 24 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)(3,14,16,46)(4,20,23,42)(6,11,30,43)(7,13,29,44)(8,32,33,15)(9,37,39,24)(12,31,45,22)(17,38,52,62)(18,47,27,55)(19,48,51,64)(21,34,50,59)(25,41,28,56)(26,40,54,63)(35,58,61,53)(36,57,60,49);(1,3,7,19)(2,8,13,36)(4,21,26,53)(5,27,29,52)(6,18,22,17)(9,38,41,47)(10,33,44,60)(11,35,12,34)(14,20,48,40)(15,25,49,24)(16,30,51,31)(23,32,54,57)(28,50,37,58)(39,55,56,62)(42,46,63,64)(43,61,45,59)

Polycyclic presentation:

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

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1855);