#6583205

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574084
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2996 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(5,6)(7,19)(8,15)(9,14)(10,18)(11,17)(12,22)(13,27)(16,33)(20,26)(21,31)(23,29)(24,34)(25,32)(28,35)(30,36);(5,7,10,22,11,23,33,24)(6,12,16,19,17,34,18,29)(8,25,26,21,27,36,28,14)(9,20,30,15,31,35,32,13);(1,2,3,4)(5,8)(6,13)(7,14)(9,29)(10,26)(11,27)(12,30)(15,17)(16,20)(18,35)(19,31)(21,23)(22,25)(24,36)(28,33)(32,34);(5,9)(6,14)(7,20)(8,12)(10,30)(11,31)(13,24)(15,22)(16,25)(17,21)(18,36)(19,26)(23,35)(27,34)(28,29)(32,33);(5,8)(6,15)(7,21)(9,29)(10,26)(11,27)(12,30)(13,17)(14,23)(16,35)(18,20)(19,31)(22,36)(24,25)(28,33)(32,34)

Polycyclic presentation:

2^9|b^2=f,c^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6490401);