#397150

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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

870 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,5,8)(3,7,4,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,33)(21,39)(22,27)(23,28)(25,36)(26,31)(29,40)(30,47)(32,45)(34,43)(35,41)(37,46)(38,44)(42,48);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(17,19,22,21)(18,24,27,39)(20,34,36,48)(23,32,40,31)(25,42,33,43)(26,28,45,29)(30,38,35,37)(41,46,47,44);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(17,20)(18,25)(19,30)(21,35)(22,36)(23,38)(24,41)(26,43)(27,33)(28,46)(29,44)(31,48)(32,34)(37,40)(39,47)(42,45);(1,4,5,3)(2,6,8,7)(17,21,22,19)(18,26,27,45)(20,35,36,30)(23,32,40,31)(24,28,39,29)(25,43,33,42)(34,37,48,38)(41,46,47,44)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 304346);