#507597

order	512
nilpotency class	4
isoclinism	isoclinic class of #500307
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	#29273
prime radical	C₂³	#575
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

399 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 399 conjugacy classes of subgroups — too many to draw. Only its 142 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)(3,15,19,28)(4,20)(5,9)(6,23)(7,12)(8,14)(10,18,29,16)(11,22)(13,21)(17,31,25,30)(24,27,32,26);(2,9,14,12)(4,21)(6,22)(10,27,29,26)(11,20,23,13)(15,31,28,30)(16,18)(24,32);(1,3)(2,10,14,29)(4,16)(5,17)(6,24)(7,25)(8,19)(9,26,12,27)(11,28,23,15)(13,30,20,31)(18,21)(22,32);(1,4,5,22,8,21,7,6)(2,11,12,20,14,23,9,13)(3,16,17,32,19,18,25,24)(10,28,27,31,29,15,26,30)

Polycyclic presentation:

2^9|a^2=eh,b^2=g,c^2=g,d^2=e,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,d^b=di,f^b=fi,f^c=fh,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, 414793);