#6465275

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1337 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=f,d^2=hi,e^2=fgi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,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, 6372471);