#413682

order	512
nilpotency class	3
isoclinism	isoclinic class of #395166
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

760 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ei,c^2=e,d^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320878);