#415472

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

938 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)(9,10,12,14)(11,15,16,13)(17,18)(19,29)(20,34)(21,28)(22,27)(23,26)(24,31)(25,36)(30,42)(32,40)(33,41)(35,39)(37,44)(38,43)(45,46)(47,48);(1,3)(2,5)(4,7)(6,8)(17,19,21,31,23,33,22,32)(18,24,27,29,26,40,28,41)(20,35,36,47,37,30,38,46)(25,42,34,48,43,39,44,45);(3,7)(5,8)(17,20)(18,25)(19,30)(21,36)(22,38)(23,37)(24,39)(26,43)(27,34)(28,44)(29,45)(31,46)(32,47)(33,35)(40,42)(41,48);(9,11,12,16)(10,13,14,15)(18,26)(24,40)(25,43)(27,28)(29,41)(34,44)(39,42)(45,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=e,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, 322668);