#112526

order	512
nilpotency class	3
isoclinism	isoclinic class of #112521
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1010 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1010 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,d^b=di,g^b=ghi,d^c=di,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19722);