#6524287

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500456
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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1444 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:

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

Polycyclic presentation:

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

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, 6431483);