#506357

order	512
nilpotency class	4
isoclinism	isoclinic class of #500596
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 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 537 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:

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

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,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, 413553);