#396366

order	512
nilpotency class	3
isoclinism	isoclinic class of #395163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1405 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=i,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^c=fi,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, 303562);