#94426

order	512
nilpotency class	5
isoclinism	isoclinic class of #94423
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7304
commutator	C₂²×D₄	C₄²
Frattini, cosocle	C₂×#536	C₂²
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×D₄ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#536 ◁ G
LECS	G ▷ C₂×#536 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

414 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 44 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,d^2=i,e^2=i,f^2=i,b^a=bc,c^a=cf,e^a=eg,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=dfi,h^b=hi,d^c=dhi,f^c=fi,e^d=ei,f^d=fi

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1622);