#459879

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

944 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=efi,c^2=e,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,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, 367075);