#419451

order	512
nilpotency class	3
isoclinism	isoclinic class of #419450
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
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

1421 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:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=e,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,h^c=hi,f^d=fi,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, 326647);