#108311

order	512
nilpotency class	3
isoclinism	isoclinic class of #8465
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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1020 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:

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

Polycyclic presentation:

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

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, 15507);