#107443

order	512
nilpotency class	3
isoclinism	isoclinic class of #8412
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	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

858 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14639);