#6448621

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3512	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3266 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,21)(16,30)(17,22)(18,35)(19,24)(20,23)(25,43)(26,36)(27,42)(28,38)(29,37)(31,44)(32,40)(33,39)(34,41);(2,4)(13,15,19,28)(14,21,24,38)(16,31,17,33)(18,29,20,27)(22,39,30,44)(23,42,35,37)(25,34,26,32)(36,40,43,41);(6,9)(10,12)(13,16,19,17)(14,22,24,30)(15,25,28,26)(18,32,20,34)(21,36,38,43)(23,40,35,41)(27,31,29,33)(37,44,42,39);(5,7,8,11)(6,10,9,12)(13,17,19,16)(14,22,24,30)(15,26,28,25)(18,34,20,32)(21,36,38,43)(23,40,35,41)(27,33,29,31)(37,44,42,39);(13,18,19,20)(14,23,24,35)(15,27,28,29)(16,32,17,34)(21,37,38,42)(22,40,30,41)(25,31,26,33)(36,44,43,39)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6355817);