#7519096

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519083
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

2580 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)(5,6,11,17)(7,18,8,13)(9,24,25,33)(10,16,22,15)(12,21,28,20)(14,30,32,26)(19,31,23,27)(29,35,34,36);(1,2)(3,4)(5,7,10,20,11,8,22,21)(6,12,15,18,17,28,16,13)(9,23,26,36,25,19,30,35)(14,31,24,29,32,27,33,34);(2,4)(9,25)(14,32)(19,23)(24,33)(26,30)(27,31)(29,34)(35,36);(2,4)(5,8)(6,13)(7,11)(9,23)(10,21)(12,15)(14,29)(16,28)(17,18)(19,25)(20,22)(24,27)(26,36)(30,35)(31,33)(32,34);(5,9,22,30,11,25,10,26)(6,14,16,33,17,32,15,24)(7,19,21,36,8,23,20,35)(12,27,13,29,28,31,18,34)

Polycyclic presentation:

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

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, 7426292);