#432709

order	512
nilpotency class	3
isoclinism	isoclinic class of #417464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 339905);