#404776

order	512
nilpotency class	3
isoclinism	isoclinic class of #394289
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

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

843 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)(17,18,21,24)(19,29,31,41)(20,34,35,42)(22,28,23,27)(25,33,38,32)(26,37,39,36)(30,44,45,40)(43,47,48,46);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(18,24)(25,38)(26,39)(27,28)(29,41)(34,42)(40,44)(43,48);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(17,19,23,33,21,31,22,32)(18,25,28,41,24,38,27,29)(20,30,37,47,35,45,36,46)(26,40,42,48,39,44,34,43);(9,11,12,15)(10,14,13,16)(17,20)(18,26)(19,30)(21,35)(22,36)(23,37)(24,39)(25,40)(27,34)(28,42)(29,43)(31,45)(32,46)(33,47)(38,44)(41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311972);