#448607

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

1067 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)(10,13)(11,15)(17,18,22,27,24,28,23,38)(19,25,30,41,32,42,31,47)(20,33,34,43,21,35,37,26)(29,44,45,40,36,48,46,39);(1,3,5,8)(2,6,4,7)(9,10,12,13)(11,16,15,14)(17,19)(18,25)(20,29)(21,36)(22,30)(23,31)(24,32)(26,39)(27,41)(28,42)(33,44)(34,45)(35,48)(37,46)(38,47)(40,43);(1,2,5,4)(3,6,8,7)(9,11,12,15)(10,14,13,16)(17,20,24,21)(18,26,28,43)(19,29,32,36)(22,37,23,34)(25,39,42,40)(27,35,38,33)(30,46,31,45)(41,48,47,44);(1,4,5,2)(3,6,8,7)(10,13)(14,16)(17,21,24,20)(18,26,28,43)(19,29,32,36)(22,37,23,34)(25,40,42,39)(27,33,38,35)(30,45,31,46)(41,48,47,44)

Polycyclic presentation:

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

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, 355803);