#6474092

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1356 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,28)(20,26)(21,27)(22,24)(23,37)(25,30)(29,45)(31,41)(32,48)(33,43)(34,39)(35,38)(36,44)(40,47)(42,46);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(18,24)(25,38)(26,39)(27,37)(28,41)(40,42)(43,44)(45,48);(1,3,5,4)(2,7,8,6)(11,15)(13,16)(17,19,21,35,22,31,23,30)(18,25,37,41,24,38,27,28)(20,32,36,47,34,29,33,46)(26,42,43,45,39,40,44,48);(1,4,5,3)(2,6,8,7)(11,15)(13,16)(17,20,22,34)(18,26,24,39)(19,29,31,32)(21,36,23,33)(25,40,38,42)(27,44,37,43)(28,45,41,48)(30,47,35,46);(10,14)(13,16)(17,21,22,23)(18,27,24,37)(19,30,31,35)(20,33,34,36)(25,41,38,28)(26,43,39,44)(29,46,32,47)(40,45,42,48)

Polycyclic presentation:

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

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, 6381288);