#434070

order	512
nilpotency class	3
isoclinism	isoclinic class of #419895
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

1189 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)(5,6,10,22,12,17,11,16)(7,18,20,13,21,24,14,8)(9,25,27,15,29,33,28,32)(19,30,26,34,36,23,35,31);(5,7,10,20,12,21,11,14)(6,13,16,18,17,8,22,24)(9,26,27,36,29,35,28,19)(15,31,25,23,32,34,33,30);(5,8,12,13)(6,14)(7,16)(9,23,29,30)(10,24,11,18)(15,26)(17,20)(19,33)(21,22)(25,36)(27,34,28,31)(32,35);(2,4)(5,9,12,29)(6,15,17,32)(7,19,21,36)(8,23,13,30)(10,28,11,27)(14,26,20,35)(16,33,22,25)(18,34,24,31)

Polycyclic presentation:

2^9|a^2=eg,b^2=e,c^2=efi,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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, 341266);