#152252

order	512
nilpotency class	5
isoclinism	isoclinic class of #149793
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2126
commutator	C₂×C₄²	C₂²×C₄
Frattini, prime radical, cosocle	#532	C₂³
socle	C₂	#12798
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

299 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 299 conjugacy classes of subgroups — too many to draw. Only its 68 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,c^2=h,d^2=i,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gh,h^a=hi,c^b=ci,e^c=ehi,f^c=fghi,g^c=gi,f^e=fhi,g^f=gi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59448);