#438220

order	512
nilpotency class	3
isoclinism	isoclinic class of #422131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

974 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,h^b=hi,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, 345416);