#190228

order	512
nilpotency class	3
isoclinism	isoclinic class of #190224
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

935 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=e,d^2=i,f^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,g^b=gh,f^c=fi,g^c=gh

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, 97424);