#111010

order	512
nilpotency class	3
isoclinism	isoclinic class of #9279
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

778 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 44 points:

(5,6)(7,9)(8,11)(10,12)(13,14)(15,23,26,19)(16,27)(17,30)(18,21)(20,29)(22,32)(24,38,39,44)(25,34,31,36)(28,41)(33,40,37,43)(35,42);(1,2,3,4)(6,9)(7,12)(10,11)(13,15)(14,19)(16,24)(17,31)(18,26)(20,33)(21,23)(22,36)(25,32)(27,38)(28,39)(29,43)(30,34)(35,37)(40,42)(41,44);(13,16,17,29,18,28,32,42)(14,20,22,27,21,35,30,41)(15,24,31,43,26,39,25,40)(19,33,36,38,23,37,34,44)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=i,e^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,d^b=di,e^b=ehi,f^b=fhi,d^c=dhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18206);