#180809

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂×C₄³ · Schur multiplier (H₂) = C₂³×C₄²

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

1058 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6,8,12,9,13,16,20)(7,11,15,19,14,10,17,18)(21,22)(23,26)(24,28)(25,27)(29,33)(30,32)(31,34)(35,36);(5,7)(6,10)(8,15)(9,14)(11,13)(12,18)(16,17)(19,20)(21,23,24,29,25,30,31,35)(22,26,28,33,27,32,34,36);(1,2,3,4)(5,7)(6,11)(8,15)(9,14)(10,13)(12,19)(16,17)(18,20)(22,27)(26,32)(28,34)(33,36);(7,14)(10,11)(15,17)(18,19)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,e^2=h,f^2=i,b^a=bh,c^a=ci,d^b=dh,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, 88005);