#6524635

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500186
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

1443 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)(3,6)(4,8)(5,7)(17,18,23,36,22,25,21,27)(19,28,35,24,31,41,30,38)(20,32,37,40,33,26,34,47)(29,43,48,46,44,42,45,39);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,31)(18,24,25,38)(20,29,33,44)(21,35,23,30)(26,39,32,46)(27,41,36,28)(34,48,37,45)(40,43,47,42);(2,7)(6,8)(10,13)(11,15)(18,25)(24,38)(26,32)(27,36)(28,41)(39,46)(40,47)(42,43);(9,11,12,15)(10,14,13,16)(17,20,22,33)(18,26,25,32)(19,29,31,44)(21,34,23,37)(24,39,38,46)(27,40,36,47)(28,42,41,43)(30,45,35,48);(1,4,5,3)(2,8,7,6)(10,13)(14,16)(20,33)(26,32)(29,44)(34,37)(39,46)(40,47)(42,43)(45,48)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6431831);