#121136

order	512
nilpotency class	3
isoclinism	isoclinic class of #121133
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1443
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

340 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 104 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) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28332);