#139469

order	512
nilpotency class	4
isoclinism	isoclinic class of #139440
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	#2619	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#541 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

447 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 447 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 64

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=i,d^2=ghi,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gi,h^a=hi,d^b=dh,e^b=eghi,f^b=fhi,g^b=gi,h^b=hi,d^c=dghi,e^c=ei,f^c=fghi,f^d=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46665);