#6524690

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

1441 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)(17,18,21,26,22,25,35,40)(19,27,29,37,30,41,34,23)(20,31,33,47,32,24,38,39)(28,43,45,36,44,42,48,46);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,23,25,37)(20,28,32,44)(21,34,35,29)(24,36,31,46)(26,41,40,27)(33,48,38,45)(39,43,47,42);(2,7)(6,8)(10,13)(11,15);(9,11,12,15)(10,14,13,16)(17,20,22,32)(18,24,25,31)(19,28,30,44)(21,33,35,38)(23,36,37,46)(26,39,40,47)(27,42,41,43)(29,45,34,48);(1,4,5,3)(2,8,7,6)(10,13)(14,16)(18,25)(20,32)(23,37)(26,40)(27,41)(28,44)(33,38)(45,48)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,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, 6431886);