#148024

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,d^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, 55220);