#134152

order	512
nilpotency class	4
isoclinism	isoclinic class of #133885
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	C₂×#537	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#125 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 714 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) = 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)(19,23)(20,30,31,45)(21,33,35,25)(22,26)(24,32)(27,41,42,44)(28,40,34,37)(29,38)(36,43)(39,46)(47,48);(1,3,6,13)(2,7,9,11)(4,12,5,14)(8,15,10,16)(17,19,22,29)(18,23,26,38)(20,27,31,42)(21,34,35,28)(24,36,39,48)(25,40,33,37)(30,44,45,41)(32,47,46,43);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(17,20)(18,24,26,39)(19,27)(21,32)(22,31)(23,36,38,48)(25,30,33,45)(28,43)(29,42)(34,47)(35,46)(37,41,40,44)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=gh,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,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, 41348);