#148030

order	512
nilpotency class	4
isoclinism	isoclinic class of #148027
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11043
commutator	C₂³×C₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

757 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 757 conjugacy classes of subgroups — too many to draw. Only its 132 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fi,h^a=hi,c^b=cf,f^b=fi,d^c=di,e^c=ei,g^c=ghi,g^e=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55226);