#507766

order	512
nilpotency class	4
isoclinism	isoclinic class of #500306
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3227
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#29392
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

419 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 419 conjugacy classes of subgroups — too many to draw. Only its 124 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=e,d^2=g,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,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, 414962);