#6579965

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1494 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:

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

Polycyclic presentation:

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

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, 6487161);