#134125

order	512
nilpotency class	4
isoclinism	isoclinic class of #133843
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#544 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

669 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 669 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=i,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=di,e^b=ei,f^b=fgi,d^c=di,e^c=ei,f^c=fhi,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41321);