#184376

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

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

646 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=eg,d^2=ef,f^2=h,g^2=i,b^a=be,d^a=di,c^b=ch

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91572);