#139709

order	512
nilpotency class	4
isoclinism	isoclinic class of #139709
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1889
commutator	C₂²×Q₈	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	#23909	C₂
LCS	G ▷ C₂²×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

695 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 695 conjugacy classes of subgroups — too many to draw. Only its 68 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)(3,5,7,8)(4,6)(9,10,14,19)(11,20,24,38)(12,25,21,17)(13,18)(15,30,33,22)(16,28)(23,37)(26,35)(27,36)(29,34)(31,40)(32,39);(1,3)(2,5,6,8)(4,7)(9,11,21,30,14,24,12,22)(10,15,25,38,19,33,17,20)(13,27,29,39,28,23,26,40)(16,31,34,37,18,32,35,36);(10,16)(13,28)(15,31)(17,35)(18,19)(20,36)(23,27)(25,34)(26,29)(32,33)(37,38)(39,40)

Polycyclic presentation:

2^9|a^2=f,b^2=df,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,h^a=hi,d^b=dgi,e^b=ehi,f^b=fgi,d^c=dh,e^c=ei,f^c=fh,e^d=ei,f^e=fi

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, 46905);