#94443

order	512
nilpotency class	5
isoclinism	isoclinic class of #94441
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7307
commutator	C₂×C₄²	C₄²
Frattini, cosocle	C₂×#536	C₂²
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#536 ◁ G
LECS	G ▷ C₂×#536 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 426 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=hi,d^2=i,f^2=i,b^a=bc,c^a=cf,e^a=eghi,f^a=fh,h^a=hi,c^b=cg,d^b=dfh,g^b=gi,d^c=dhi,e^c=ei,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, 1639);