#499658

order	512
nilpotency class	4
isoclinism	isoclinic class of #499263
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3150	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

700 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406854);