#6441439

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1838 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fgi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6348635);