#152691

order	512
nilpotency class	5
isoclinism	isoclinic class of #149894
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2122
commutator	C₂×#41	C₂²×C₄
Frattini, cosocle	#532	C₂³
socle	C₂	#13083
prime radical	#23994	C₂
LCS	G ▷ C₂×#41 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3444 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

473 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 473 conjugacy classes of subgroups — too many to draw. Only its 64 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=h,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,c^b=ci,d^b=dg,f^b=fg,g^b=gh,f^d=fhi,g^d=gi,g^f=gi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59887);