#131737

order	512
nilpotency class	4
isoclinism	isoclinic class of #131668
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	#62340	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

583 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 583 conjugacy classes of subgroups — too many to draw. Only its 90 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)(17,18,22,25,24,30,20,36,23,29,37,43,21,28,39,27)(19,31,34,47,33,46,32,42,35,48,41,44,40,26,38,45);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(18,25)(20,37)(21,24)(22,39)(26,42)(27,30)(28,36)(29,43)(31,47)(32,41)(33,40)(34,38)(44,48)(45,46);(2,8)(4,5)(7,15)(9,10)(11,16)(12,14)(17,19)(18,26)(20,38)(21,40)(22,41)(23,35)(24,33)(25,42)(27,47)(28,48)(29,46)(30,31)(32,39)(34,37)(36,44)(43,45)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38933);