#503121

order	512
nilpotency class	4
isoclinism	isoclinic class of #501060
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1301 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 1301 conjugacy classes of subgroups — too many to draw. Only its 190 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=fh,d^2=e,f^2=hi,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,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410317);