#121258

order	512
nilpotency class	3
isoclinism	isoclinic class of #121257
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

363 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 363 conjugacy classes of subgroups — too many to draw. Only its 108 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28454);