#422613

order	512
nilpotency class	3
isoclinism	isoclinic class of #419794
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

990 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:

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

Polycyclic presentation:

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

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, 329809);