#138987

order	512
nilpotency class	4
isoclinism	isoclinic class of #138850
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1752 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

340 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 74 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)(3,6,10,13)(4,7)(5,8)(9,12,15,16)(11,14)(17,18,23,29,24,30,21,28,25,31,39,47,38,46,22,40)(19,27,33,26,34,43,20,36,35,42,37,44,41,48,32,45);(1,3,5,10)(2,6)(4,9,11,15)(7,12)(8,13)(14,16)(17,19,24,41,25,35,38,34)(18,26,30,45,31,44,46,36)(20,21,33,39,32,22,37,23)(27,40,48,29,42,28,43,47);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,12,13,16)(17,20)(18,27)(19,21)(22,35)(23,34)(24,33)(25,32)(26,40)(28,44)(29,45)(30,48)(31,42)(36,47)(37,38)(39,41)(43,46)

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,c^2=g,d^2=ghi,e^2=ghi,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=eghi,f^b=fh,h^b=hi,d^c=dghi,e^c=egh,f^c=fghi,h^c=hi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46183);