#505682

order	512
nilpotency class	4
isoclinism	isoclinic class of #500112
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

544 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 154 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=e,c^2=i,d^2=e,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fi,f^c=fhi,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, 412878);