#6589717

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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×#447	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1638 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 6496913);