#131713

order	512
nilpotency class	4
isoclinism	isoclinic class of #131648
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#62252	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

576 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 94 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=hi,e^2=g,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,f^b=fgh,h^b=hi,e^c=eg,f^c=fg

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38909);