#121926

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	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

365 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 365 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(2,7)(3,11)(4,5)(8,16)(12,15)(13,14)(17,18,22,27,23,26,21,35)(19,28,30,39,31,24,37,40)(20,32,34,25,33,43,36,42)(29,44,47,38,45,48,46,41);(1,2,4,10,6,7,5,9)(3,12,11,8,13,16,14,15)(17,19,21,30,23,31,22,37)(18,24,35,40,26,28,27,39)(20,29,36,47,33,45,34,46)(25,41,32,44,42,38,43,48);(1,3,4,11,6,13,5,14)(2,8,10,16,7,15,9,12)(17,20,21,36,23,33,22,34)(18,25,35,32,26,42,27,43)(19,29,30,47,31,45,37,46)(24,38,40,48,28,41,39,44)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,e^2=hi,f^2=i,g^2=h,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, 29122);