#110725

order	512
nilpotency class	3
isoclinism	isoclinic class of #9331
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

696 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17921);