#441200

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
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

1281 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)(9,10,12,14)(11,13,15,16)(17,18,23,36,21,24,22,27)(19,28,35,25,30,41,31,38)(20,32,37,39,33,42,34,26)(29,44,47,43,45,40,46,48);(1,3,5,8)(2,6,4,7)(9,11,12,15)(10,13,14,16)(18,24)(25,38)(26,39)(27,36)(28,41)(32,42)(40,44)(43,48);(1,2,5,4)(3,6,8,7)(11,15)(13,16)(17,19,21,30)(18,25,24,38)(20,29,33,45)(22,35,23,31)(26,40,39,44)(27,41,36,28)(32,43,42,48)(34,47,37,46);(1,4,5,2)(3,6,8,7)(17,20)(18,26)(19,29)(21,33)(22,34)(23,37)(24,39)(25,40)(27,42)(28,43)(30,45)(31,46)(32,36)(35,47)(38,44)(41,48)

Polycyclic presentation:

2^9|a^2=efg,b^2=egi,c^2=e,d^2=ei,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, 348396);