#455977

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

1150 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 363173);