#180841

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂×C₄³ · Schur multiplier (H₂) = C₂×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄²	C₂⁴
commutator	C₂²	C₂×C₄³
socle	C₂³	C₂²×C₄²
prime radical	C₂⁴	C₂×C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1116 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,e^2=h,f^2=i,c^a=ch,d^b=di,d^c=dh

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88037);