#112664

order	512
nilpotency class	3
isoclinism	isoclinic class of #10128
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

798 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

(1,2,6,10,5,11,16,9)(3,7,13,14,12,4,8,15)(17,18)(19,26,28,23)(20,24)(21,25)(22,27,32,29)(30,31);(1,3,5,12)(2,7,11,4)(6,8,16,13)(9,14,10,15)(17,19)(18,22)(20,29)(21,28)(23,24)(25,32)(26,31)(27,30);(1,4)(2,8)(3,10)(5,7)(6,15)(9,12)(11,13)(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,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,d^b=di,f^b=fh,g^b=ghi,d^c=dhi,f^c=fh,g^c=gi

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, 19860);