#131698

order	512
nilpotency class	4
isoclinism	isoclinic class of #131664
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	#62487	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 536 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=hi,e^2=gi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,f^b=fgh,h^b=hi,e^c=egi,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, 38894);