#163252

order	512
nilpotency class	3
isoclinism	isoclinic class of #163012
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	C₂×#525	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1143 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cgh,d^a=dgi,e^a=eg,c^b=ci,e^b=eh,f^b=fgh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341053, #72341054, #72341058
maximal normal subgroup of	#72341053, #72341054, #72341058

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 70448);