#434077

order	512
nilpotency class	3
isoclinism	isoclinic class of #419919
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1164 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)(5,6,11,15,10,17,9,16)(7,18,25,14,21,27,20,12)(8,13,22,31,23,30,24,29)(19,33,34,28,36,32,35,26);(5,7,9,20,10,21,11,25)(6,12,15,18,17,14,16,27)(8,19,22,34,23,36,24,35)(13,28,29,33,30,26,31,32);(5,8,9,22,10,23,11,24)(6,13,16,31,17,30,15,29)(7,19,25,35,21,36,20,34)(12,26,18,32,14,28,27,33);(2,4)(5,7)(6,14)(8,19)(9,25)(10,21)(11,20)(12,17)(13,28)(15,18)(16,27)(22,35)(23,36)(24,34)(26,30)(29,32)(31,33)

Polycyclic presentation:

2^9|a^2=eg,b^2=e,c^2=efi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,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, 341273);