#116630

order	512
nilpotency class	3
isoclinism	isoclinic class of #116629
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

578 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 130 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)(3,9)(4,12)(5,8)(6,10)(7,13)(11,16)(14,15)(17,18)(19,29)(20,33,37,43)(21,27)(22,26)(23,28)(24,31)(25,36,44,35)(30,45,47,39)(32,40)(34,48,46,42)(38,41);(4,13)(6,15)(7,12)(10,14)(17,19,21,31,23,32,22,38)(18,24,26,29,28,41,27,40)(20,34,35,30,37,46,36,47)(25,42,33,45,44,48,43,39);(1,3,5,11)(2,6,8,15)(4,14,13,10)(7,16,12,9)(17,20)(18,25)(19,30)(21,35)(22,36)(23,37)(24,39)(26,33)(27,43)(28,44)(29,42)(31,46)(32,47)(34,38)(40,48)(41,45)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,d^2=h,e^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,e^b=ei,f^b=fh,e^c=ehi,f^c=fh,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, 23826);