#120980

order	512
nilpotency class	3
isoclinism	isoclinic class of #120978
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1442
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#125
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 369 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28176);