#416850

order	512
nilpotency class	3
isoclinism	isoclinic class of #394189
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

772 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=e,d^2=hi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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, 324046);