#152775

order	512
nilpotency class	5
isoclinism	isoclinic class of #152774
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13090
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#532	C₂³
prime radical	#3498	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #3498 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

523 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 523 conjugacy classes of subgroups — too many to draw. Only its 66 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)(3,14,16,31)(4,20,6,12)(5,11)(7,13)(8,17,27,18)(9,21,10,23)(15,30)(19,26,24,25)(22,29,32,28);(1,3,15,18,7,19,5,22)(2,8,9,25,13,29,10,31)(4,16,21,32,6,24,23,17)(11,14,20,27,30,26,12,28);(1,4,7,6)(2,9,13,10)(3,16,19,24)(5,23,15,21)(8,25,29,31)(11,12,30,20)(14,28,26,27)(17,18,32,22)

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=i,d^2=h,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dgh,e^b=ei,f^b=fghi,g^b=ghi,h^b=hi,d^c=di,e^c=ei,f^c=fi,f^d=fhi,g^f=gi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59971);