#458583

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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

891 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=f,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, 365779);