#451036

order	512
nilpotency class	3
isoclinism	isoclinic class of #417542
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

1486 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)(12,23,25,16)(13,17,26,32)(14,18)(20,29)(21,28)(24,30)(27,34)(31,35)(33,39,40,38)(36,37);(2,5)(3,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,29)(17,31)(18,19)(23,27)(24,33)(25,34)(26,35)(28,32)(30,38)(36,39)(37,40);(9,12)(10,16)(11,20)(13,24)(14,25)(15,27)(17,30)(18,23)(19,29)(21,33)(22,34)(26,37)(28,38)(31,39)(32,36)(35,40);(1,3,4,7)(2,6,5,8)(9,13,14,26)(10,17)(11,21,22,35)(12,24,25,37)(15,28)(16,30)(18,32)(19,31)(20,33,34,40)(23,36)(27,38)(29,39)

Polycyclic presentation:

2^9|a^2=gh,d^2=ei,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^b=hi,h^c=hi,f^d=fi,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, 358232);