#6459309

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1502 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)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,21,26,22,27,33,23)(19,24,28,38,30,39,43,35)(20,31,32,36,34,25,40,41)(29,44,42,46,45,37,47,48);(2,6)(7,8)(18,23)(19,28)(20,32)(21,33)(26,27)(29,45)(30,43)(34,40)(35,38)(36,41)(37,48)(44,46);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,19,21,28,22,30,33,43)(18,24,26,38,27,39,23,35)(20,29,32,42,34,45,40,47)(25,37,41,48,31,44,36,46);(1,4,5,3)(2,8,6,7)(17,20,33,40,22,34,21,32)(18,25,23,36,27,31,26,41)(19,29,43,47,30,45,28,42)(24,37,35,46,39,44,38,48);(9,11,12,15)(10,14,13,16)(20,34)(25,31)(29,45)(32,40)(36,41)(37,44)(42,47)(46,48)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6366505);