#499655

order	512
nilpotency class	4
isoclinism	isoclinic class of #498863
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#493
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#23798
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

695 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 695 conjugacy classes of subgroups — too many to draw. Only its 160 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:

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

Polycyclic presentation:

2^9|a^2=e,d^2=efh,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,d^b=di,e^b=eh,f^b=fh,h^b=hi,e^c=ehi,f^c=fhi,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, 406851);