#135555

order	512
nilpotency class	4
isoclinism	isoclinic class of #135554
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8749
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

419 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 419 conjugacy classes of subgroups — too many to draw. Only its 60 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=dh,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,e^b=ei,f^b=fgi,d^c=di,e^c=eg,f^c=fh,h^c=hi,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, 42751);