#512074

order	512
nilpotency class	4
isoclinism	isoclinic class of #512074
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32979
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#32945	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

944 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 172 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:

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

Polycyclic presentation:

2^9|c^2=eh,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,h^a=hi,f^b=fi,e^c=ei,f^c=fhi,h^c=hi,g^d=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419270);