#138503

order	512
nilpotency class	4
isoclinism	isoclinic class of #138488
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#2018	C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2922 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

346 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 78 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)(3,9)(4,12,14,7)(5,11)(6,8)(10,13,16,15)(17,18,20,28,23,27,37,46,22,29,36,26,21,38,35,45)(19,30,31,40,32,25,34,24,33,41,47,43,39,44,48,42);(1,3,9,2,6,11,5,8)(4,10,13,7,14,16,15,12)(17,19)(18,24,29,42)(20,34)(21,32)(22,33)(23,39)(25,28,44,26)(27,40,38,43)(30,46,41,45)(31,37)(35,47)(36,48);(1,4)(2,7)(3,10)(5,15)(6,14)(8,12)(9,13)(11,16)(17,19)(18,25)(20,31)(21,39)(22,33)(23,32)(24,28)(26,42)(27,41)(29,44)(30,38)(34,37)(35,48)(36,47)(40,45)(43,46)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,d^2=ghi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=egi,f^b=fh,h^b=hi,d^c=dgi,e^c=ei,f^c=fgi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45699);