#131616

order	512
nilpotency class	4
isoclinism	isoclinic class of #131224
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂×C₄⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	#392	D₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

621 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 621 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gh,d^2=h,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dh,e^b=eg,f^b=fgh,d^c=dgi,f^c=fg,e^d=ei,f^d=fi,f^e=fi

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, 38812);