#109406

order	512
nilpotency class	3
isoclinism	isoclinic class of #8674
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#380
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

614 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 16602);