#163389

order	512
nilpotency class	3
isoclinism	isoclinic class of #162550
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1134 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1134 conjugacy classes of subgroups — too many to draw. Only its 234 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)(3,7,8,5)(4,6)(9,10)(11,19,21,24)(12,16)(13,18)(14,17)(15,20,23,22)(25,26,29,33,30,34,28,32)(27,35,38,31,36,40,37,39);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,20)(13,22)(14,21)(16,19)(17,23)(18,24)(25,27)(26,31)(28,38)(29,37)(30,36)(32,40)(33,35)(34,39);(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,19,23,24)(27,36)(31,39)(35,40)(37,38);(2,6)(5,7)(10,17)(15,23)(16,18)(19,24)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=hi,e^2=h,f^2=i,b^a=be,d^a=dghi,e^a=eg,c^b=ci,e^b=eh,f^b=fgh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 70585);