#94350

order	512
nilpotency class	5
isoclinism	isoclinic class of #7422
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1347
commutator	C₂×C₁₆	C₄²
Frattini, cosocle	C₂²×M₃₂	C₂²
socle	C₂	#7258
prime radical	C₂⁴	Q₃₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂²×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

214 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fh,d^2=i,e^2=i,f^2=hi,g^2=hi,h^2=i,b^a=bc,c^a=cf,e^a=efgh,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=ghi,h^b=hi,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1546);