#411181

order	512
nilpotency class	3
isoclinism	isoclinic class of #395169
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

926 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)(3,6)(4,5)(7,8)(11,15)(13,16)(17,18,22,28,23,29,40,27)(19,30,21,24,32,26,39,42)(20,33,37,45,38,46,36,25)(31,43,48,44,34,41,35,47);(2,5)(6,8)(9,10,12,14)(11,16,15,13)(17,19,22,21,23,32,40,39)(18,24,27,30,29,42,28,26)(20,34,36,48,38,31,37,35)(25,43,33,44,45,41,46,47);(9,11,12,15)(10,13,14,16)(17,20,23,38)(18,25,29,45)(19,31,32,34)(21,35,39,48)(22,36,40,37)(24,41,42,43)(26,44,30,47)(27,33,28,46);(1,3,4,7)(2,6,5,8)(17,21,22,32,23,39,40,19)(18,26,27,24,29,30,28,42)(20,35,36,34,38,48,37,31)(25,44,33,41,45,47,46,43)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318377);