#506500

order	512
nilpotency class	4
isoclinism	isoclinic class of #501332
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2190	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 789 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) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eg,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,c^b=cg,d^b=di,f^c=fh,h^c=hi,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413696);