#137155

order	512
nilpotency class	4
isoclinism	isoclinic class of #137153
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#2018	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

741 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 741 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,d^2=g,e^2=i,f^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ehi,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,e^c=ei,f^c=fh

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, 44351);