#416821

order	512
nilpotency class	3
isoclinism	isoclinic class of #394189
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

894 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)(12,13)(14,16)(17,18)(19,29)(20,34)(21,28)(22,27)(23,26)(24,31)(25,35)(30,44)(32,39)(33,40)(36,42)(37,41)(38,46)(43,45)(47,48);(1,3)(2,5)(4,7)(6,8)(10,13)(14,15)(17,19,22,32,23,33,21,31)(18,24,28,40,26,39,27,29)(20,30,37,47,36,46,35,45)(25,38,42,48,41,44,34,43);(3,7)(5,8)(17,20)(18,25)(19,30)(21,35)(22,37)(23,36)(24,38)(26,41)(27,34)(28,42)(29,43)(31,45)(32,47)(33,46)(39,44)(40,48);(9,11,12,16)(10,14,13,15)(18,26)(24,39)(25,41)(27,28)(29,40)(34,42)(38,44)(43,48)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=hi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,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, 324017);