#121851

order	512
nilpotency class	3
isoclinism	isoclinic class of #121851
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1445
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

390 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 116 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29047);