#145524

order	512
nilpotency class	4
isoclinism	isoclinic class of #145505
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2054
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3169	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

262 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 60 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)(3,10)(4,12)(5,13)(6,9)(7,11)(8,15)(14,16)(17,18,24,32)(19,33,38,47)(20,25,40,28)(21,29,39,27)(22,31,23,30)(26,41,42,35)(34,45,48,46)(36,43,37,44);(1,3)(2,7)(4,8)(5,14)(6,12)(9,16)(10,13)(11,15)(18,25)(19,34)(20,39)(21,40)(22,23)(26,42)(27,30)(28,32)(29,31)(33,44)(35,37)(36,41)(38,48)(43,47);(1,4,5,7,6,10,15,16)(2,3,8,11,9,12,13,14)(17,19,21,41,22,36,20,34,24,38,39,35,23,37,40,48)(18,26,28,47,30,45,27,43,32,42,25,33,31,46,29,44)

Polycyclic presentation:

2^9|a^2=i,c^2=dh,d^2=gi,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=egi,f^b=fhi,g^b=gi,h^b=hi,e^c=eghi,f^c=fgh

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, 52720);