#427426

order	512
nilpotency class	3
isoclinism	isoclinic class of #425091
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1273 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,g^d=gi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 334622);