#6463784

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1972 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)(13,14,19,24)(15,25,27,37)(16,30,32,40)(17,23,18,22)(20,29,36,28)(21,33,39,34)(26,35,31,38)(41,42,44,43);(2,4)(5,6)(7,11)(8,9)(10,12)(13,15,18,29,19,27,17,28)(14,20,23,37,24,36,22,25)(16,31,34,42,32,26,33,43)(21,38,30,41,39,35,40,44);(6,9)(10,11)(13,16,18,34,19,32,17,33)(14,21,23,30,24,39,22,40)(15,26,29,43,27,31,28,42)(20,35,37,44,36,38,25,41);(5,7,8,12)(6,10,9,11)(15,27)(20,36)(25,37)(26,31)(28,29)(35,38)(41,44)(42,43);(16,32)(21,39)(26,31)(30,40)(33,34)(35,38)(41,44)(42,43)

Polycyclic presentation:

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

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, 6370980);