#401936

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

1200 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)(5,6,11,14,10,15,9,23)(7,16,24,27,17,30,19,12)(8,20,25,28,21,34,22,13)(18,32,35,36,31,26,33,29);(5,7,9,19,10,17,11,24)(6,12,14,16,15,27,23,30)(8,18,22,33,21,31,25,35)(13,26,20,29,28,32,34,36);(7,17)(8,21)(13,28)(19,24)(20,34)(22,25)(26,32)(29,36);(2,4)(5,8,9,22,10,21,11,25)(6,13,14,20,15,28,23,34)(7,18,19,33,17,31,24,35)(12,26,16,29,27,32,30,36)

Polycyclic presentation:

2^9|a^2=eg,b^2=e,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=di

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, 309132);