#506359

order	512
nilpotency class	4
isoclinism	isoclinic class of #500599
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

539 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 539 conjugacy classes of subgroups — too many to draw. Only its 144 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:

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

Polycyclic presentation:

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

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, 413555);