#6454241

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1962 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)(15,20)(16,21)(17,22)(18,24)(19,23)(25,35)(26,36)(27,38)(28,37)(29,39)(30,40)(31,41)(32,42)(33,44)(34,43);(2,4)(13,15,19,27)(14,20,23,38)(16,29,17,31)(18,33,34,28)(21,39,22,41)(24,44,43,37)(25,32,26,30)(35,42,36,40);(6,9)(10,12)(13,16,19,17)(14,21,23,22)(15,25,27,26)(18,32,34,30)(20,35,38,36)(24,42,43,40)(28,31,33,29)(37,41,44,39);(13,17,19,16)(14,22,23,21)(15,26,27,25)(18,30,34,32)(20,36,38,35)(24,40,43,42)(28,29,33,31)(37,39,44,41);(5,7,8,11)(6,10,9,12)(14,23)(15,27)(21,22)(24,43)(25,26)(28,33)(29,31)(40,42)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=i,e^2=g,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, 6361437);