#130438

order	512
nilpotency class	4
isoclinism	isoclinic class of #130254
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#62327	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

575 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 575 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 128

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

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

Polycyclic presentation:

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

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, 37634);