#7519072

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519063
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

2572 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)(5,6,12,18)(7,19,9,15)(8,14,11,17)(10,25,27,31)(13,20,29,22)(16,23,32,26)(21,35,24,28)(30,36,33,34);(1,2)(3,4)(5,7,11,22,12,9,8,20)(6,13,14,19,18,29,17,15)(10,21,26,36,27,24,23,34)(16,28,25,33,32,35,31,30);(2,4)(5,8,12,11)(6,14,18,17)(7,20,9,22)(10,23,27,26)(13,19,29,15)(16,25,32,31)(21,34,24,36)(28,33,35,30);(2,4)(5,9)(6,15)(7,12)(8,22)(10,24)(11,20)(13,14)(16,30)(17,29)(18,19)(21,27)(23,36)(25,28)(26,34)(31,35)(32,33);(5,10,8,23,12,27,11,26)(6,16,17,31,18,32,14,25)(7,21,20,34,9,24,22,36)(13,28,15,30,29,35,19,33)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,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,h^b=hi,h^d=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, 7426268);