#232710

order	512
nilpotency class	3
isoclinism	isoclinic class of #232706
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1276 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 1276 conjugacy classes of subgroups — too many to draw. Only its 204 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 — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=di,e^b=eh,g^b=gh,e^c=ei,f^c=fi,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139906);