#134070

order	512
nilpotency class	4
isoclinism	isoclinic class of #133835
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

733 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 733 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 64

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,e^b=ei,f^b=fg,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, 41266);