#7514193

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

3142 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)(8,13)(9,14)(10,17)(11,18)(12,21)(15,26)(16,27)(20,33)(22,35)(23,29)(24,31)(25,32)(28,34)(30,36);(1,2)(3,4)(5,7)(6,12)(8,22)(9,18)(10,14)(11,15)(13,23)(16,26)(17,19)(20,25)(21,27)(24,34)(28,32)(29,36)(30,33)(31,35);(2,4)(5,8,11,25)(6,13,18,32)(7,20,15,22)(9,23,12,28)(10,24,27,36)(14,29,21,34)(16,30,17,31)(19,33,26,35);(2,4)(5,9,11,12)(6,14,18,21)(7,17,15,16)(8,23,25,28)(10,26,27,19)(13,29,32,34)(20,31,22,30)(24,35,36,33);(5,9)(6,15)(7,18)(8,23)(10,19)(11,12)(13,22)(14,17)(16,21)(20,32)(24,33)(25,28)(26,27)(29,31)(30,34)(35,36)

Polycyclic presentation:

2^9|c^2=i,d^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,e^d=ei,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, 7421389);