#106605

order	512
nilpotency class	3
isoclinism	isoclinic class of #8346
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

958 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:

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

Polycyclic presentation:

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

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, 13801);