#466254

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×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

887 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:

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

Polycyclic presentation:

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

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, 373450);