#121928

order	512
nilpotency class	3
isoclinism	isoclinic class of #121925
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	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

354 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 112 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:

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

Polycyclic presentation:

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

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, 29124);