#7514175

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514172
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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 ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3117 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:

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

Polycyclic presentation:

2^9|c^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cgi,d^b=dh,f^b=fi,h^b=hi,h^d=hi,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7421371);