#423845

order	512
nilpotency class	3
isoclinism	isoclinic class of #422131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1268 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=f,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,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, 331041);