#6443408

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1970 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=g,d^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,e^c=ei,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, 6350604);