#112608

order	512
nilpotency class	3
isoclinism	isoclinic class of #112605
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

876 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19804);