#152362

order	512
nilpotency class	5
isoclinism	isoclinic class of #149799
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2128
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#532	C₂³
socle	C₂	#12859
prime radical	#3455	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3455 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

364 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 60 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)(3,14,4,18)(5,10,22,31)(7,13,15,29)(8,20,27,24)(9,16,28,19)(12,25,32,23)(17,26,21,30);(1,3)(2,8,13,28)(4,15)(5,16)(6,21)(7,17)(9,29,27,11)(10,14,12,26)(18,31,30,32)(19,25)(20,22)(23,24);(1,4,5,19,7,21,23,20)(2,9,10,30,13,27,12,18)(3,15,16,25,17,6,24,22)(8,11,14,31,28,29,26,32)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=d,d^2=i,e^2=h,g^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ch,e^b=ei,f^b=fi,e^c=eh,f^c=fghi,g^c=gi,f^e=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, 59558);