#451025

order	512
nilpotency class	3
isoclinism	isoclinic class of #417538
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1483 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358221);