#121127

order	512
nilpotency class	3
isoclinism	isoclinic class of #121125
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	#117
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

356 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 106 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)(3,10,12,6)(4,7)(5,8)(9,13)(11,15,16,14)(17,18,21,26,22,25,36,41)(19,27,30,44,29,23,37,39)(20,31,34,42,33,24,35,40)(28,43,45,48,32,38,46,47);(1,3,4,11,5,12,13,16)(2,6,7,14,8,10,9,15)(17,19)(18,23,25,27)(20,32)(21,30)(22,29)(24,38,31,43)(26,39,41,44)(28,33)(34,46)(35,45)(36,37)(40,47,42,48);(2,7,8,9)(6,14,10,15)(17,20)(18,24)(19,28,29,32)(21,35)(22,33)(23,38,27,43)(25,31)(26,42)(30,46,37,45)(34,36)(39,48,44,47)(40,41)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,e^2=h,g^2=i,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,f^c=fi,g^c=gi

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, 28323);