#151872

order	512
nilpotency class	5
isoclinism	isoclinic class of #13605
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2127
commutator	C₄²	C₂×C₄²
Frattini, cosocle	#532	C₂³
socle	C₂	#12582
prime radical	#3449	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #3449 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

563 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 563 conjugacy classes of subgroups — too many to draw. Only its 120 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=h,e^2=i,g^2=i,b^a=bd,d^a=dg,f^a=fh,g^a=gh,h^a=hi,c^b=ch,e^b=eghi,d^c=di,e^d=ehi,f^e=fi,g^e=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59068);