#461655

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

985 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=f,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,h^c=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, 368851);