#506297

order	512
nilpotency class	4
isoclinism	isoclinic class of #499953
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3205
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#28156
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 639 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)(3,14)(4,12)(5,8)(6,22)(7,13)(9,15)(10,20)(11,21)(16,29)(17,30)(18,27)(19,25)(23,26)(24,32)(28,31);(2,8,13,11)(4,19)(6,20)(9,26,28,24)(10,12,22,25)(14,29,27,30)(15,31)(23,32);(1,3)(2,9)(4,15)(5,16)(6,23)(7,18)(8,24)(10,27)(11,26)(12,29)(13,28)(14,22)(17,21)(19,31)(20,32)(25,30);(1,4,5,20,7,19,21,6)(2,10,11,12,13,22,8,25)(3,15,16,32,18,31,17,23)(9,27,26,29,28,14,24,30)

Polycyclic presentation:

2^9|b^2=g,d^2=e,e^2=i,f^2=hi,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, 413493);