#434825

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1000 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 342021);