#466529

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
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

765 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)(19,29)(20,25)(21,27)(22,28)(23,38)(24,32)(26,36)(30,40)(31,47)(33,45)(34,41)(35,43)(37,44)(39,46)(42,48);(2,5)(6,7)(9,11,13,12)(10,14,15,16)(17,19,23,34,21,32,22,33)(18,24,38,45,27,29,28,41)(20,31,37,30,35,48,36,46)(25,42,44,39,43,47,26,40);(1,3,4,8)(2,6,5,7)(10,15)(14,16)(17,20,21,35)(18,25,27,43)(19,30,32,46)(22,37,23,36)(24,39,29,40)(26,28,44,38)(31,33,48,34)(41,47,45,42);(3,8)(6,7)(9,12,13,11)(10,16,15,14)(17,20)(18,26)(19,31)(21,35)(22,36)(23,37)(24,40)(25,38)(27,44)(28,43)(29,39)(30,34)(32,48)(33,46)(41,47)(42,45)

Polycyclic presentation:

2^9|a^2=f,b^2=efg,c^2=e,d^2=efi,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, 373725);