#506355

order	512
nilpotency class	4
isoclinism	isoclinic class of #500307
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3227
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#28202
prime radical	#3150	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 521 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)(3,15,18,30)(4,19)(5,9)(6,22)(7,12)(8,14)(10,16,27,31)(11,21)(13,20)(17,28,24,29)(23,25,32,26);(2,9,14,12)(4,20)(6,21)(10,26,27,25)(11,19,22,13)(15,28,30,29)(16,31)(23,32);(1,3)(2,10)(4,16)(5,17)(6,23)(7,24)(8,18)(9,25)(11,15)(12,26)(13,28)(14,27)(19,29)(20,31)(21,32)(22,30);(1,4,5,21,8,20,7,6)(2,11,12,19,14,22,9,13)(3,16,17,32,18,31,24,23)(10,15,26,29,27,30,25,28)

Polycyclic presentation:

2^9|a^2=eh,b^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=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, 413551);