#506406

order	512
nilpotency class	4
isoclinism	isoclinic class of #500591
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3253
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#28251
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

678 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 678 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413602);