#466981

order	512
nilpotency class	3
isoclinism	isoclinic class of #457096
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

992 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=gi,c^2=ef,d^2=fi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=di,d^c=de,e^c=ei,f^c=fi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 374177);