#386309

order	512
nilpotency class	3
isoclinism	isoclinic class of #219532
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

700 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=g,d^2=ei,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 293505);