#6445196

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439035
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#24289	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1942 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|c^2=g,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6352392);