#148026

order	512
nilpotency class	4
isoclinism	isoclinic class of #148023
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

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

normal subgroups only

This group has 765 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,d^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,g^c=gh,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, 55222);